The Swedenborgian or "New Church" denominations that accept the Christian theology and Bible interpretations of Emanuel Swedenborg (1688–1772) interpret Matthew 24:5 and similar verses in precisely this way: as referring, not to people claiming to be Christ, but to Christian churches and people teaching false doctrine as Christian truth, and in this way deceiving people and leading them astray.
Passages from Apocalypse Explained
Here are three statements to that effect from Swedenborg's unpublished (by him) commentary on the book of Revelation, Apocalypse Explained:
"Many shall come in my name, saying, I am: go ye not therefore after them." (Luke 21:8; Mark 13:6)
These people "coming in the name of the Lord" and saying "I am" denotes preaching false doctrines and declaring that they are true, and thus leading astray. The same is signified by these words in Matthew:
"Many shall come in my name, saying, I am Christ, and shall deceive many." (Matthew 24:5, 11, 23-27)
For by "Jesus" is meant the Lord as to Divine good, and by "Christ," the Lord as to Divine truth (Arcana Coelestia, #3004, 3005, 3009, 5502), and by not being Christ, is signified, not Divine truth, but falsity. (Apocalypse Explained #102:3, links added for references)
And another (where Swedenborg refers to the Bible as "the Word," which was the standard term for it in his day). Here he states very plainly that this passage does not mean people calling themselves Christ:
"See that no one seduce you; for many shall come in my name, saying, I am the Christ, and shall seduce many. If any one say to you, Lo, here is Christ, or there, believe it not, for there will arise false Christs and false prophets." (Matthew 24:4, 5, 23, 24; Mark 13:21-23)
This must not be understood as meaning that there will arise those who will call themselves the Christ or Christs, but those who will falsify the Word, and say that this or that is Divine Truth when it is not. Those who confirm falsities from the Word are meant by false Christs, and those who propagate falsities of doctrine are meant by false prophets. For these two chapters treat of the successive devastation of the church, thus of the falsification of the Word, and at length of the profanation of truth therefrom. But these things may be seen further explained in the Arcana Coelestia, #3353–3356, and #3897–3901. (Apocalypse Explained #684:7, links added for the first section in each range of references)
And another, where by "church" Swedenborg means the existing Christian Church:
"Many shall come in my name, saying, I am Christ, and shall lead many astray. But ye shall hear of wars and rumors of wars; see that ye be not troubled; for nation shall rise up against nation and kingdom against kingdom, and there shall be famines, and pestilences, and earthquakes." (Matt. 24:5–7; Mark 13:6–8; Luke 21:8–11)
This was said by the Lord to the disciples concerning the consummation of the age, which signifies the state of the church at its end, which is described in those chapters, therefore it also means the successive perversion and falsification of the truth and good of the Word, until nothing remains but falsity and evil therefrom. Those who shall come in His name and call themselves Christ, and shall lead many astray, signify that those shall come who will say that this is Divine Truth, when nevertheless it is truth falsified, which in itself is falsity; for by Christ is meant the Lord as to Divine Truth, but here, in the opposite sense, truth falsified. (Apocalypse Explained #734:24)
In other words, Swedenborg is saying that at the consummation of the age, by which he understood the spiritual end of the existing Christian Church, Christians and the various Christian churches would teach false doctrine as Christian truth, and that this is the meaning of "false Christs."
Passages from Arcana Coelestia
In Arcana Coelestia ("Secrets of Heaven"), Swedenborg provides a detailed commentary on the "Little Apocalypse" in Matthew 24. It is far too long to quote here, but it may be found in Arcana Coelestia #3353–3356, 3486–3489, 3650–3655, 3751–3757, 3897–3901, 4056–4060, 4229–4231, 4332–4335, and 4422–4424 (links are provided for the first section in each range).
Here are the parts of his commentary most relevant to the question.
First, he provides his overall view of the prophecies of the end times, which, he says, are not to be understood as relating to literal events that will take place physically in the material world, but rather as having a spiritual meaning relating to events that will take place spiritually in humanity in general and in the Christian Church in particular:
The majority of people believe that when the Last Judgement comes everything visible in the world is going to perish—that is to say, the earth will go up in flames, the sun and moon will be reduced to nothing, and the stars will disappear; and after that a new heaven and a new earth will come into being. They have acquired this idea from the prophetical revelations, among which such occurrences are mentioned. But what will in fact happen at that time is quite different, as becomes clear from what has been shown already concerning the Last Judgement in 900, 931, 1850, 2117-2133. Those paragraphs show that the Last Judgement is nothing else than the end of the Church with one group of people and the beginning of it with another. This end with one and beginning with another occurs when the Lord is not acknowledged any longer, or what amounts to the same, when there is no faith any longer. No acknowledgement or faith exists any longer when there is no charity any longer, for faith is in no way possible except with those in whom charity is present. In those circumstances the Church comes to an end and is transferred to others, as is plainly evident from all the things that the Lord Himself taught and foretold in the Gospels concerning the last day or the close of the age—in Matthew 24; Mark 13; and Luke 21. But since nobody without the key, which is the internal sense, is able to understand those things foretold by Him there, let them be explained one after another. (Arcana Coelestia #3353:1, links added)
He then goes on to provide a verse-by-verse explanation of the entire chapter, starting with this:
Here first let the following words be explained which appear in Matthew,
The disciples came to Jesus, saying, Tell us. when will those things take place, and what will be the sign of Your coming and of the close of the age? And Jesus answering said to them, See that no one leads you astray, for many will come in My name, saying, I am the Christ; and they will lead many astray. But when you hear of wars and rumors of wars, see that you are not alarmed; for all things must take place; but the end is not yet. For nation will be roused against nation and kingdom against kingdom, and there will be famines, and plagues, and earthquakes in various places. All these are but the beginning of sorrows. (Matthew 24:3–8)
Those who confine themselves to the sense of the letter cannot know whether these words and those that follow in this chapter refer to the destruction of Jerusalem and the dispersion of the Jews, or whether they refer to the end of days which is called the Last Judgement. But those admitted into the internal sense see clearly that the end of the Church is being referred to, this end being that which here and in other places is called "the coming of the Lord" and "the close of the age." And inasmuch as the end of the Church is meant one is able to see that all these statements made by the Lord mean such things as have to do with the Church. But their overall meaning may be seen from the individual meaning below which each of them has in the internal sense.
Many will come in My name, saying, I am the Christ; and they will lead many astray.
"Name" here does not mean name, nor "the Christ" the Christ, but "name" means that by which the Lord is worshipped, 2724, 3006, while "the Christ" means truth itself, 3009, 3010. Thus the meaning is that people will come who say that this is the sum and substance of faith, that is, it is the truth, when in fact it is neither the sum and substance of faith, nor the truth, but falsity. (Arcana Coelestia #3353:2, links added)
I have not quoted the rest of his explanation of the initial section of Matthew quoted, but you can see it at the final link provided.
In the next section, he steps back and comments on how the Christian Church began to become corrupt, and its teachings falsified:
These individual meanings show what the Lord's words are used to mean, namely the first state of the perversion of the Church, which occurs when people cease to know any longer what good is and what truth is, and instead argue with one another about them, which gives rise to falsities. But because this is only the first state it is said that the end is not yet, and that these are the beginning of sorrows; and that state is referred to as earthquakes in various places, which in the internal sense means an alteration of the state of the Church–a partial or initial alteration. The fact that this was told to the disciples means that it was addressed to all who belonged to the Church, for the twelve disciples represented these, 2089, 2129, 2130. This explains why they are told, "See that no one leads you astray," and also, "When you hear of wars and rumors of wars, see that you are not alarmed." (Arcana Coelestia #3354, links added)
When he speaks of this occurring "when people cease to know any longer what good is and what truth is, and instead argue with one another about them, which gives rise to falsities," he is referring to the division, discord, rancorous debate about doctrine within Christianity that led up to and followed the First Council of Nicaea in 325 AD—which he saw as heralding the beginning of the end of the Christian Church both doctrinally and in terms of Christian charity.
The existing Christian Church as "false Christs"
Swedenborg, then, saw the "false Christs" as coming very early in the history of the existing Christian church, when various Christian theologians and bishops began arguing about doctrinal tenets instead of following the two Great Commandments given by Jesus Christ himself: to love the Lord our God with all our heart, soul, and mind, and to love our neighbor as ourselves (Matthew 22:34–40; Mark 12:28–34; Luke 10:25–28)—which had been the spirit and practice of the early Christians of the first century or so after Christ.
When, instead of loving one another as Christ commanded, the various leaders and parties within Christianity began attacking one another, anathematizing one another, excommunicating one another, exiling one another, and damning one another to hell, this, Swedenborg believed, marked the effective end of the Christian Church as a truly Christian church that followed the teachings of Jesus Christ as given in the Gospels. And the subsequent dark history of the institutions of Christianity and their internecine battles only added more evidence to Swedenborg's view that starting in at least 325 AD, the Christian church was "Christian in name only, but not in reality and essence" (True Christian Religion #668).
The resulting doctrines that came out of the First Council of Nicaea and the subsequent councils and creeds over the centuries, right up through the Protestant Reformation and its further degradation and falsification of Christian doctrine, Swedenborg believed, represented "false Christs" that deceived and led astray the entire Christian Church throughout the vast bulk of its history.
Summary and conclusion
I recognize that this view will not sit will with members of the existing Christian Church.
However, this serves to show that Emanuel Swedenborg and the various denominations that accept his Christian theology and Bible interpretations do indeed understand Matthew 25:4, "For many will come in my name, saying, 'I am the Christ' and they will lead many astray," as referring, not to people claiming to be Christ, but rather as people who are believers in Christ (meaning Christians) using Christ's good name to deceive many by teaching false doctrines.
I would simply ask that if you vote on this answer, you vote not based on whether or not you agree with this answer or even find the beliefs expressed troubling, but on whether it provides a solid answer to the question asked.
Whether or not you agree with Swedenborg's reasoning, conclusions, and views of the existing Christian Church in all of its major branches, he did teach precisely the alternative view of Matthew 24:5 that the question asks about—and that view is held to in the Swedenborgian or New Church denominations that follow his teachings.