They won't turn around and give it to Republicans - even though they should - because that's a bridge too far ideologically for them.

No, they shouldn't. For the last seven or eight decades, the GOP has been the home of those who advocate that the US is a uniquely Christian nation and those who support a worldview that modern advocates call "heritage Americans." That America isn't a creedal country based on ideas or pluralism, but instead is a nation that is characterized by the particular people who live here or a specific culture. The whole, "western civilization is the best!" thing. That there's a certain cultural background whose members have made the US a success, and those who are outside that cultural background generally have not.

That type of approach to a society tends to turn out badly for small discrete minority groups that aren't historically part of the religious majority or 'heritage' ethnic group. We also don't do well in "populist" political movements that frame issues as one where the righteous People are in battle against a corrupt Elite - because we usually get lumped in with the corrupt Elite. That latter dynamic is what you see taking root in certain parts of the Democratic party, but it doesn't make the Republican party any more hospitable - especially given the rise of right-wing populism here and abroad.