I only know enough to comment on Christianity, not religion globally.
My personal opinion on Paul's letters is that he was writing and commenting on common social practices in Greece and Rome 2000 years ago because that is, in fact, what he was doing.
Paul was focused on establishing and strengthening the new church. His concerns over sexual practices in ancient Rome have little to nothing to do with, say, gay marriage. The modern concept of homosexuality did not exist at that time.
The church, so far as I know, had no official stance on homosexuality until Saint Thomas Aquinas decided to systematize church law by throwing Paul and Aristotle into a blender. I believe this was approximately 1,200 years after Paul's death. This was a fateful decision with long-lasting ramifications, even if contemporary fundamentalists know nothing about ancient history, Aquinas, or Aristotle.
The Christian religion is not, by nature, homophobic. One must, in fact, make many leaps of logic to assert that it is inherently so. Such beliefs were bolted-on after the fact, and it remains so to this day.
Do bigots use Christianity to rationalize their bigotry? Absolutely. Neither one causes the other. They have simply been engaged in a poorly conceived, albeit effective, partnership.