That is a solvable issue -- give examples of manipulation and misinformation for both sides in roughly equal quantities. There is no shortage of examples for any argument.
Nothing helps making people manipulation resistant like seeing manipulation from the side of the argument they support.
We currently have separate narratives of "<side 1> engages in manipulation, here is their misinformation" which polarizes and causes highly emotional reactions against the evil supporters of that side. This also means that people, due to that emotional response, are unwilling to consider opposite views and we get the polarized society we have today.
If instead we are teaching with examples of manipulation of both sides of the same argument, people see the value of debating on merits, instead of emotions, cross checking references and correcting for the speaker's viewpoint. They also get less affected by misinformation, since they have seen it, many times, including coming from the side near and dear to their hearts.
This last thing is critical. One would not get too riled up about messaging if they see both sides doing that. My 2c.
I was taught that multiple times under sections on political cartoons/propaganda, at least when I went to school.
The issue is not that people are not well educated, its that there is not enough reason for individuals at a large scale to invest time deep-diving into a topic when they can use other things like social cues and popularity to more or less decide where they fit on the spectrum.
The issue is not that manipulation is invisible to the untrained eye, on the contrary I'd argue a lot more people are aware of political manipulation than you realize. It doesn't take much logic to see politicians only start caring about you once elections come around, regardless of where you fall on the political slider.
People are optimizing for social harmony and the security of their livelihood. This means that more or less whatever the opinions of those in my community / work / place of worship are, I will reciprocate those signals to some degree or another unless there is an issue that resonates strongly enough. Insert mass-scale reach with social media and the law of large numbers in human population and I hope you can see why there is no non-technological solution to this problem.
We have created a dopamine machine in our pocket capable of satisfying the natural urges of every permutation of the mind. Your bigger picture and moral appeal is given a moment of consideration by an ever-shrinking percentage.
Nothing helps making people manipulation resistant like seeing manipulation from the side of the argument they support.