The sun will exist for hundreds of thousands of years after humanity has gone extinct. The sun will exist for millions of years before it burns out.

Your timescales are off. Even if humanity lasts a very long time, which seems unlikely, the sun will last for billions of years after humanity is gone. In one billion years the sun will have become hotter so that life becomes impossible on Earth. There will be four billion years of a lifeless Earth before the sun expands into a red giant and either swallows up or cooks the Earth. One billion years after that the sun will kick off its outer layers into a nebula and become a white dwarf. At that point it’s not reacting any more so it just gradually cools down over billions more years until it’s just a cool lump.