Certainly, the Catholic Church had its elites too. But that set up a dual truth doctrine because the elites were the literate ones who dealt with their texts, while the message for the masses was tainted with all kinds of compromise with folklore and political agendas. Catholic theology was broadened to encompass as many foreign religious ideas as possible. Hence, Mary was venerated to encompass goddess worship, while the saints and angels were used to encompass polytheism. The more mystical and philosophical Eastern theology was purer, as opposed to being cheapened by salesmanship and imperialism. Again, I could be wrong about that, but that's my impression.

It would be crazy not to assume that the Catholic Church under Christendom was highly influential on all aspects of Western culture. Yuval Harari makes that point about empires. Empires devour and influence everything. So obviously Western art and science arose from a Christian culture. However, the fact of that influence shouldn't be confused with a value judgment. The benefits of that influence might be accidental, or modernity might have arisen faster without the Church's dominance. It's like the question of whether Ronald Reagan ended the Cold War or prolonged it (because Gorbachev wanted to end it sooner).

I tend to think that religion in some form is inevitable, if we think of religion in existential terms (as Tillich did), as being about our ultimate values or our trust in something held to be relatively sacred. Even "secularists" are religious in that sense. There are civil religions propping up values of patriotism, liberty, wealth, and so on. Maybe the question isn't so much whether politics should be mixed with religion, but which religion should be mixed with late-modern politics.