I completely agree with all the comment that point out that it does not necessarily have to do with religion or political views, especially broad terms like left and right. At the end of the day there is no detailed fixed definition for any of those terms that people use to identify themselves as.
Additionally I want to add that even member of the LGBTQ community have very different perspectives when they say they want to support it.
For instance some gay, bi etc people are very much against pride parades for instance (at least in my country), for various reasons, most of which "scream" that they simply seek acceptance through supporting a more um "serious" demeanor, as if to be accepted you have to be solemn and not stand out in any way that other people might find "flamboyant" or not serious enough.
I remember my first time witnessing a pride by complete accident, I felt so relieved. I was about 21-22 deep in the closet, seriously believing that gay people can simply just sit home and watch porn and nothing else (yes I was dumb).
Being flamboyant or being a drag queen dancing at the front row of the parade does not mean in any way that you are doing a disservice to the "movement". In fact, I would wager that promoting a more heteronormative image as the more serious one is pretty much internalised homophobia, because it leads to judging people who are just like you simply based on how they express themselves.
