Nobody said the core cools as it shrinks. You just made that up. The hydrogen fuel is exhausted, so gravitational collapse heats the core until helium fusion takes over. There is no helium or light element fusion going on right now because the sun's core is not hot enough.
Indeed, formation of a planetary nebula is very similar to a supernova. The difference is that in a supernova, the core collapses much more violently because it is heavier. Our sun is small, so it will fizzle instead of boom.



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