Most people care about having true beliefs when it actually lets them achieve things.

Here I have a general feeling that any true belief may be useful in the future, and any false belief may be harmful in the future. I feel the world as connected. (As a most obvious example, a belief in supernatural in any area implies a belief in supernatural in general, which in turn influences all areas of life.)

Maybe "the world is connected" is one of the unspoken premises for rationality. If you don't have it, any rationality technique will be merely something you use inside the lab.

(Of course, not everything is equally likely to be useful, so I try to get more info in some areas and ignore other areas. But I would still feel bad about making false beliefs even in the less important areas. If I don't feel certain about my knowledge somewhere, and don't have time to improve the knowledge, I update to "don't know".)