Culture wars are a subtle thing to wage, because they determine everything without being about anything. Explicitly political contests are at least ostensibly about some particular concrete thing: you're fighting for or against a specific law or a specific candidate. But how do you fight a narrative, when your enemy is less of a regime and more of a meme? How do you explain to anyone what you're trying to accomplish when you're not trying to get anyone to do anything different in particular, but to renounce their distorted way of thinking and speaking, after which you expect them to make better decisions, even if you can't say in advance what those decisions will be?
Picture me rushing into a room. "People, people! The standard map is wrong! Look at this way better map I found in the literature; let's use this one!"
"Our map isn't wrong. It has all the same continents yours does."
"I mean, yes, but it's a Mercator projection. Surely you don't really think Antarctica is larger than Asia?"
"Why do you care what size Antarctica is? What difference does it make? People are perfectly happy with Antarctica being the largest continent."
"But it's not true!"
"It sounds like you're assuming your beliefs are true. What is truth, anyway?"
And it being the case that no one will die if she gets the size of Antarctica wrong, what can I say to that?
I think that "after which you expect them to make better decisions" is mostly wishful thinking. In other words, as you point out at the end, the kind of changes you desire will not in fact make much difference to the world.
I think it's more honest just to admit that you care more about truth than they do. Then you can keep trying to find the truth, for yourself, and stop bothering other people about it who do not want the truth anyway, and who would not get any particular benefit from it.
Probably no need to bother with "But if I didn't assume my beliefs were true, they wouldn't actually be my beliefs!"
There's also the perspective of "what is the need to convince people?". "How great is that need?" ... is it to validate your thinking, to gain some social acceptance or some other reason?