But my question is: How do seemingly clear-headed people persist in obviously false beliefs?
If you analyze every single person who believes such a thing, you will discover that they are a part of a community that believes what they do, or were at some point a part of such a community.
It starts with a sense of belonging, which can often override reason or logical thought. It feels very good to be a part of a community, especially one that seems to answer deep questions. "Nobody else has figured this out yet, but we have!" probably makes you feel pretty good too, especially if you see others in your community reveling in that feeling as well.
If I grew up in a community of young earth creationists, I would probably be one too. It's easy to say that they are "obviously false", even if they are. If you grow up with your parents, teachers, role models, etc. believing in something, it will be a lot easier to embrace it yourself. There is always some level of peer pressure to "be one of us" as well. A lot of these communities shun members who disagree, such behaviour is usually heavily discouraged. Plus how could your parents, teachers, and other role models all be wrong? To admit that they are, even in the face of overwhelming evidence, might be too much for a lot of people. It would be admitting that not only your entire life has been a lie, but also the lives of your parents, and other people who you look up to or used to look up to. It's far easier to just go with the program and continue the belief system and structure.
It's also important to note that these beliefs are often at the very centre of the community in question. It isn't just some random belief, but usually a belief that forms one of the main pillars of the community itself. To go against the belief would be to go against the community, and in many cases the community is the only thing these people know. If you can't come home at the end of a long frustrating day, where else can you go? A person without a community is a scary proposition for a lot of people. So they hold on to the community and the core beliefs that helps it survive.