I'm an outsider on both counts - I've never belonged to a political party or been truly involved in organized religion, so maybe this should be taken with a grain of salt:
I think you have it backwards. The "religious right" has become somewhat emblematic of Christianity in the USA. If you look at any type of media, that faction is dominant. They have been integrating themselves into the Republican party since at least 1980, demanding and getting planks in the Republican party platform hostile to gays and opposing abortion.
I moved to a pretty red area a couple of years ago. I used to listen to Christian and right wing talk shows a good bit when I was in the car. They rule a lot of the airwaves here. I was astonished when, last summer, one by one they started pumping Trump for president. Let's be real. Trump is no Christian. He can't even convincingly pay Christianity lip service:
https://www.yahoo.com/news/oh-god-donald-trump-doesn-142821578.html
The religious right just doesn't care. This is about power.
Here are a couple of quotes from fundamentalist Christian radio personalities. I don't have the names of the people who made these statements or links, but they sure stuck in my head:
"Sometimes God uses bad people to do good things."
"If you would vote for a Democrat, I'm sorry, but I would have to question your salvation."
It seems to me that at least this powerful, vocal subset of Christianity is openly hostile to Democrats, as well as gays and basically anyone who doesn't toe their lines.