"Corals are more likely to thrive in warmer climates rather than colder climates."
you are correct however that is as much because of how the ocean chemistry, in this case, carbonate balance shifts with latitude.
Carbonate ion is what hard corals use to produce their skeletons, as you move further away from the equator there is less of it so they cannot grow as fast and can be outcompeted by other organisms.
You are correct, they have moved to accommodate for rising and falling sea levels, however it is important to note this happened at an extremely gradual pace over 1000s of years.
In contrast, the rapid shift we are seeing today doesn't allow for natural adaptation.
