There is a pattern that it seems to me rationalist-type people are prone to:
Examples:
I think this is error-prone because in step 2 of the journey, we found out barely anyone actually believes X, and people just say it for whatever reason. Of course, that doesn't automatically mean X is false. But when looking at self-justifications by groups that fit this pattern, they mostly lean on the wide endorsement of the belief and basically assume it as background.
But this won't convince those who are not convinced, and most people who verbally agree with X are already used to living with the contradiction. You also shouldn't lean on the wide endorsement for yourself - I guess it's some positive evidence, but considering the wide dis-endorsement in practice, it's rather weak to negative on net.
Instead, I advise that before proceeding to step 3, you independently derive the belief X - for yourself and for others you want to convince, which turns out to be almost everyone.