Yes, this is an excellent point. People rarely try to do the work of rationally weighing the evidence presented to them, and instead take shortcuts like accepting the version that seems more favorable, or that's more in line with their existing world view, or that is presented by someone they trust.
I still believe that this is something that can be nudged through education, but I think we would need a class that's more focused essentially on applied logic rather than just explaining pure logic. It's also something that would have to be consistently emphasized throughout the curriculum.
However, even more important than changing things through education would be to create the institutions necessary for thought-out debate for the general population. The internet that exists today is a better medium for this than television or radio ever were, but it's still absolutely horrible compared to in-person meetings. Most people are extremely disengaged when consuming content online, picking up only a few phrases and ideas here and there, instead of thinking rigorously and intensely on what is being said. This is inherent to a medium that allows you to browse while taking the subway, when your attention is usually low.
But that's assuming that people really will want to know what is going on, rather than that they will do their damnest to believe the things that are most convenient to them, and that will make all your carefully prepared evidence moot. It won't matter at all how right you are, you will lose the argument.