Christianity preserved the classical heritage of Europe, yes. However, the Celtic monastaries in which so many of the old texts were saved were independent of Rome. The church at that time was not the monarchical top-down institution that Roman Catholicism became centuries later. More like the geographically-based Christianity that the early Church of England divines advocated.
My point is that political freedom developed in countries where the Reformation took root. You answer, "So what?" As someone who values liberty and the Anglo-American heritage of free institutions, I have to respond, "BIG what."
Political freedom has a lot of antecedents. Some of them were Catholic, some were Protestant.
The people who wrote the Magna Carta were Catholics. So were the Swiss who founded the first modern republic in the Middle Ages. So were the English exiles in Maryland who founded the first town to practice religious toleration in the New World (a few months before Roger Williams in Rhode Island). Some aspects of the political thought of the American revolution can be traced back through John Locke (a Protestant) to Thomas More (a Catholic).
Christianity in the Dark Ages was the Catholic Church. There was no other. Whether the monasteries operated independently of Rome does not mean that they were not Catholic. If you were a Christian, you were Catholic. The Church was one until 1054 when the Orthodox split.
See also Pyro7480’s post 73 which will remind you that rights and freedoms began in Catholic England with the Magna Carta in 1215.