Education is no match for the power of motivated thinking.
A lot of critical thinking failures are trivial. If people can't see though them trivially, there's not a lot of ways to help. People fall for really obvious scams and parrot things they read without a moment's consideration.
The people spreading misinformation don't have to work very hard. There are people who would be taken in by cleverly-disguised falsehoods, but in a lot of cases they're not disguised at all. They're just outright, obvious lies, and it seems to work just as well.
Sure, I'd love to see schools do a better job than they do with critical thinking skills. We could all benefit from it. But it won't beat out basic tribalism and people's insistence on believing things that affirm their membership.