I'm not sure teaching it in school is actually going to help. Most people will tell you that of course you need to look at primary sources to verify claims - and then turn around and believe the first thing they here from LLM, Redditor, Wiki article, etc. Even worse, many people get openly hostile to the idea that people should verify claims - "what, you don't believe me?"/"everyone here has been telling you this is true, do you have any evidence it isn't?"/"oh, so you think you know better?"

There was a recent discussion about Wikipedia here recently where a lot of people who are active on the site argued against people taking the claims there with a grain of salt and verifying the accuracy for themselves.

We can teach these things until the cows come home, but it's not going to make a difference if people say it's a good idea and then immediately do the opposite.