Sounds like it is the LOB that possibly cares about the dive log, or at least that is what matters practically speaking.
If the person has a card from a recognized agency, sounds like they do, all good there.
If the person has enough dives to make the LOB happy, but no log, reconstruct one as best they can. It wouldn't be lying and wouldn't hurt to have something. People log in all sorts of different ways, some don't log at all, people rarely get them signed. No reason to get bent out of shape about the past here. But, find out what the LOB actually wants. Because really there is no such thing as proof, and I'm sure they know that. They also know dive count often doesn't have much relationship to competence, Asking someone to fill out and sign something to cover the dive op, followed by them deciding for themselves if someone can dive (and deciding what if anything to do about what they see) is pretty common.
If the person is basically a good diver but missing some skills or knowledge they should have, it would be good to fill in those areas through reading, a class, whatever. If they don't want to, and the OP isn't happy with the situation, then they shouldn't dive with them. Simple.
I agree but was responding to mac66 asking why OP cares about his buddy logging.