Nobody's claiming that all coral is going to go extinct.. the reef environment that has existed for the past few thousand years is at great risk though. Water temperatures that we know have been relatively stable for several hundred years are suddenly rapidly warming. Bleaching events due to high temps which infrequently occurred in the past are happening nearly every year now, which gives the reef no time to rejuvenate between them. The evolutionary process which protects species in their niches takes hundreds or thousands of generations to adapt to new selection pressures and the changes are happening over dozens of generations instead, which may be too fast for most species to respond.

Coral and coral reefs will surely exist for the next few hundred million years but e.g. the Great Barrier Reef as an example of a vibrant reef ecosystem might not. We don't know exactly where the tipping point for these extremely complex systems lies, but we know that it's some point in the direction we're heading and we're starting to see examples of the outcomes that scientists predict to see near those tipping points.

My guess is as a species it will relocate to somewhere with the right temperature zone but because coral takes so long to grow from the perspective of those of us alive the existing “old growth” coral will die.

0.9 C warmer on average vs location specific volatility + acidification

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