If this worked the problem would already be solved.
Assuming we can somehow defer peoples personal responsibility on logic and reasoning and _not_ have it be exploited / corrupted in the future is a hell of an assumption. It feels to me on par with the Government asking for backdoors to encryption.
It definitely is not working now but that's not a supporting argument for the impossibility of an educated population. It's an argument against the current education infrastructure. Which, by most accounts, is flawed - some would say entirely broken.
However, to your point, it's not a class and it's not required. So participation in debate ends up being very low among the population of all students.
Simply let some time pass people will get better at detecting bs once they've encounter too many lies. Same way this generation is better than the previous one.
A lot of the conspiracy theorists who have at least average intelligence become that way because they recognize the inconsistencies in mainstream views. Having to deal with the world as it is, and with imperfect information, seems to be difficult for some of the most logical people.
I think it would be better to expose people to more facts, more of history, more of science, so they get a better understanding of the range of what's likely to be true. Beyond that, people need to be exposed to people from other cultures, other groups so that it becomes harder to believe rumors about them or see them as some monolithic entity.
I mean, just look at the gender wage gap. Even the president (Obama at the time) misrepresented the wage gap at the state of the union and almost all of the media went along with it. The wage gap has been demonstrated by the BLS study (fact), but was being misused to influence to population to believe that men make about 20% more than women in the same job, all else equal.