Having grown up in the rural deep south, this feels like more of a description of blue state Republicans than red state Republicans.
You can clearly see that plenty of states have Republicans that care about LGBT issues. In Kentucky, Mississippi, Tennessee, and West Virginia, less than 1/3rd of Republicans think homosexuality should be accepted. And it’s not like all the other states are blowing them out of the water. A majority of Republicans/Republican leaners believe gay marriage being legal is somewhat or very bad for society. Less than half believe it is acceptable for gay people to raise children. And that’s not even touching trans issues.