Let's set the stage by reviewing how Catholic Answers (a popular Catholic apologist website) answers the claim: https://www.catholic.com/magazine/online-edition/how-to-answer-the-charge-catholicism-is-pagan . In the article Mr. Karl Keating (founder and senior fellow of Catholic Answer) proposes that the presence of what used to be heathen / pagan elements were introduced to the church as inculturation, a strategy defended by many popes as late as Pope St. John Paul II. Probably aware of where the criticism came from, Mr. Keating also pointed out how the same inculturation process is happening within the Protestant churches as well. In both cases, the Christian faith itself is not compromised.
What's uncontroversial was how local churches underground having no legal status within the Roman Empire until the Edict of Milan (313 AD) having to tolerate pagan worship with "state sponsored" rich festivals, sculptures, temples, rituals, etc. Money simply poured in mostly from rich senatorial members as well as local town leaders. But in the first centuries following Constantine (reigning 306-324) until the Justinian (reigning 527-565), the now legal Church started to win state protection and preferential treatment in the major cities until paganism became irrelevant and relegated to the countryside by the end of Justinian rule. I recommend https://www.thegreatcourses.com/courses/fall-of-the-pagans-and-the-origins-of-medieval-christianity.html to learn about the details.
Imagine you are the Emperor, also having the title Pontifex Maximus, the chief high priest of the Roman religion, having power to distribute offices to all institutions related to religion. Shortly after Constantine you'll have to treat both religions equal before the law and you have to manage powerful senators and city leaders across your empire politically so they keep giving you support. Remember how your predecessors have used emperor sponsorship of temples as your political base throughout the empire. At the same time, you want to promote Christianity by directing money to the building of churches, most notably the Hagia Sophia in Constantinople. When there are communities where the Christians slowly had the upper hand over the pagans you would support breaking down the temples and literally use the marbles and stones to construct a church, or you support architectural renovation to convert the structure into a church.
Now imagine you are a typical bishop between 306-565 leading your local diocese, given more funding and legal rights to do more in winning converts to Christianity. Most of your target audience would be pagans. How would you win them over? You basically have to fuse Judeo + Early Christian approach to God (heavy in morality but very few sanctioned rituals like baptism, communion, marriage) with Pagan approach (no morality but heavy in rituals). Two completely different spiritualities: one is expressed through justice, works of mercy, and legal standing before God, the other is expressed through doing the right rituals so NOT to offend the gods (otherwise calamities will fall on you, which is why Christians were persecuted in the first place, because of FEAR that enemies will destroy Rome, which is why the early Christian apologists were busy defending the Christians in not worshiping those idols). But in morals you are free to do whatever you like otherwise; you have your option of Stoic, Epicurean, Platonic philosophies for your moral code. As bishop you have to transition your pagan audience to be faithful morally and spiritually to the one God Jesus.
Then imagine you are a missionary TODAY being sent to Korea / Japan where there are STILL entrenched shaman practices and their shrines. I was told how Samsung executives TODAY have to participate in company sponsored sea-side rituals led by shaman to offer sacrifices and prayers to some gods for blessing for Samsung, with some minority Christians having to either excuse themselves or participate with their fingers crossed offering prayer to the Christian God for forgiveness for participating because they cannot afford to be fired from the job.
So as bishop how would you wean both the Roman pagans of the 4-6th centuries and the Korean+Japanese "heathens" of today from their emotional, intellectual, and superstitious attachments to elements of their paganism, so entrenched in their survival and existential instincts so you could achieve enduring non-disruptive conversion to the Christian faith? That would be the proper context to evaluate the inculturation strategy which of course in the modern West today is completely irrelevant (but WARNING: there is a rise of those old elements coming back under different names: interest in magic like Harry Potter, new age, etc.) So why not CO-OPT the popular festivals into celebrating Jesus instead?
For more details, I suggest reviewing the work of a Roman Catholic priest, later appointed as history professor at St. Thomas College in Minnesota, offered a post as professor of reformation history at the University of Notre Dame, and awarded the title of monsignor, Mgr. Philip Hughes, who wrote the popular and very readable "A History of the Church to the Eve of the Reformation (first published in 1936, revised 1976) in 3 Volumes. The first volume (-711) is the most relevant, with these chapters heavily address paganism:
- Chapter 1. The World in which the Church was Founded. Chapter Sections: The Roman Imperial Unity, The Pagan Religions of the Romano-Hellenistic Culture, The Religion of the Jews, Tendencies in the Religious World of the First Century A.D.
- Chapter 3. The First Contacts with the Pagan Religious World. Chapter Sections: The Religious World of the Second Century, The First Apologists, The Gnostics and the Church, St. Irenaeus of Lyons, Marcion -- Montanism
- Chapter 6. The Church and the Pagan Roman Empire. Chapter Sections: The State -- Hostile and Tolerant, The State -- De-Paganised
I hope the Catholic references above will show that it's not that Christianity was paganised by the Roman empire, but instead it's a top down inculturation strategy implemented over a few centuries to win over converts and eventually knocking out the underlying pagan referents from the festivals, rituals and symbols. It's like baptism and incarnation: the old is still there, but the core killed and replaced by a new life nurtured by the Holy Spirit in the service of the true God.
St. Thomas Aquinas of the 13th century will do the same in the intellectual sphere, baptizing the newly rediscovered Aristotle into the service of Catholic theology for many centuries to come. Many Protestants would similarly protest (until today!) that there are too much Greek philosophical elements smuggled into Catholic theology, but in the end, is it really damaging to the spiritual Christian core? How about enlightenment ideas smuggled into Protestant theologies and increasingly secular Western nations today?