The doctrine of biblical inerrancy is very recent in Christian history
First, let's put biblical inerrancy into perspective.
Two centuries or so ago, and for all of Christian history before that, not a single Christian church, denomination, or preacher held that the Bible is inerrant. The very idea of biblical inerrancy had never even occurred to anyone.
It was only after the Age of Enlightenment (from the 1650s to the 1780s) that any need was seen by Christians for any such doctrine. This need was perceived only when the rapid development of science in the wake of the Age of Enlightenment began to call into question in many people's minds the scientific accuracy of many stories and statements in the Bible, such as the creation of the world in six days and the Flood of Noah.
So for roughly 1,800 years of its history, all of Christianity looked to the Bible as the Word of God without the need for any doctrine of biblical inerrancy.
Biblical inerrancy is not the same as biblical literalism
Further, biblical inerrancy is not the same thing as biblical literalism. Even the well-known 1978 Chicago Statement on Biblical Inerrancy "allows for figurative, poetic and phenomenological language, so long as it was the author's intent to present a passage as literal or symbolic" (quote from linked Wikipedia article; italics in the original).
In other words, biblical inerrancy does not require a wholly literal interpretation of the Bible.
Biblical inerrancy is not necessary for faith in the Bible
Given that all Christian churches functioned reasonably well through many centuries of Christian history without the relatively recent doctrine of biblical infallibility, the idea that now, two thousand years into Christianity, faith in God and the Bible will suddenly crumble without an adherence to biblical infallibility seems rather shaky and unfounded.
Adherence to the doctrine of biblical infallibility is by no means universal among Christian churches. However, I'll leave it to other answerers to say which denominations do and do not adhere to biblical infallibility.
Biblical infallibility misses the spiritual point of the Bible
I can, however, speak for my own denomination, the Swedenborgian Church, on the subject.
From a Swedenborgian perspective, the doctrine of biblical infallibility simply isn't very useful.
In practice, it is used mostly to assert that the Bible is without scientific and historical error. At other times it is used to assert that particular doctrines are correct because they are claimed to be based on the statements of the Bible.
However, from a Swedenborgian perspective, the Bible was never intended to be a textbook of science and history. Why would God be concerned to teach us about science and history? How would that accomplish God's purpose in speaking to humans on earth?
God's purpose is not to give us correct scientific and historical information. Rather, it is to save our souls and bring us to heaven. So what God wants to convey to humans in the Bible is not material world information, which does not affect our salvation, but spiritual understanding, which does affect our salvation.
The Bible is about spiritual change, not about correct doctrine
Further, the point of the Bible is not to instill correct doctrine. Yes, correct doctrine is important. But the purpose of doctrine in Christianity is to lead to faith in the Lord, repentance, reformation, and new spiritual life.
In other words, God's underlying purpose in the Bible is not to get us to think straight, but to get us to live a Christlike life. Faith and doctrine are merely the doorway to making us into "new creations in Christ" (see 2 Corinthians 5:17).
Biblical inerrancy is a useless doctrine for this purpose because it concerns only the head, not the heart. In the Bible, God is seeking to reach our heart, and to "replace our heart of stone with a heart of flesh" (see Ezekiel 11:19; 36:26).
The original writers of the Bible, whether or not they believed or realized they were writing under divine inspiration, all understood that the main point of their writing was to reach people with a message from God and get people to leave behind their evil and sinful lives and live good and loving lives instead. Jesus himself summed up the entirety of Scripture in these words:
"You shall love the Lord your God with all your heart, and with all
your soul, and with all your mind." This is the greatest and first
commandment. And a second is like it: "You shall love your neighbor as
yourself." On these two commandments hang all the law and the
prophets. (Matthew 22:37-40)
That is the whole purpose of Scripture according to Jesus himself: to prompt us to love the Lord our God above all else, and to love our neighbor as ourselves.
The doctrine of biblical inerrancy touches only the head.
What God wants to reach is our heart. And all of Scripture is designed by God to touch our hearts and prompt us to turn to God and live from God's love instead of from our own selfish and materialistic desires.
A more spiritual basis for Bible interpretation
Speaking for my own denomination and theology, we avoid any "slippery slope" (to the best of our finite and fallible human ability) by looking to the Bible not as a textbook of science and history, or even as a textbook of theology, but as a message from God whose purpose is to turn us toward God, faith, goodness, love, and spiritual life.
Our mode of Bible interpretation, then, does not require or rely on inerrancy. Instead, it focuses on the spiritual messages contained in and delivered by the text.