Surely it's worth asking why so many people share posts they believe to be factual so unquestioningly in the first place? Is it really so naive to think we couldn't significantly improve the average Joe's critical thinking skills? If not then it seems unlikely there's any real solution that doesn't have unwanted downsides.
But having some sort of AI capable of doing basic fact checks mightn't be so crazy... (if we can accept a red squiggly line under a misspelled word, would it bother anyone to have it automatically added under verifiably factually incorrect claims?)