The term "fact checker" does not mean someone who checks that facts are true. It means someone who checks that what someone claims is a fact is in fact a fact.
That said I agree wholeheartedly with your assessment on critical thought. It should be everyone's duty to verify facts for themselves but... consider how many people are functionally illiterate in the world. They wouldn't be able to verify even simple facts, because they lack the mental framework to do so (i.e. reading comprehension). And also consider how much of a burden it is to individually verify every single claim that is made, and compare it to how easy it is to make wild claims. It would be impossible to navigate the modern world without a group of dedicated fact checkers that you trust. You can't be an expert on everything -- and if you think you are, that means precisely that you are NOT.
The top 10% of the us population own more than 60% of the countries wealth.
In the US about a third of this disparity is due to inheritance.
That's not what I've seen, anecdotally. In school, university, and work places, the friend groups I've observed and been a part of have primarily aligned around racial groups.
Just some food for thought: How is the ratio of blue collar / white collar kids in your peer group? How close are you / were you in percentile distance?
No one caress for race in my environment, but even as a student somehow everybody I hang out with had the same amount of money.
Incidentally: A "fact" is an immutable facet of the world itself, whether we can even acknowledge it properly or not. Facts don't need checkers.