It's not that massively complex problem. Did you post on Reddit? Did you post on bitcointalk? Let's say yes. Did anyone came up with real proof that it belongs to them? There are a lot of ways to prove if you really want to and it's not only signing a message. If no one came and you waited for weeks, months, then what's wrong if you consider that from now on, it belongs to you? You haven't stolen from someone. Someone lost it, you tried your best to find the owner but owner did least to find you, then it belongs to you. You push yourself too much, you think like you are the killer if you tried to assist wounded person but failed to escape him.
I think it's a massively complex problem. I'd hate to be in this scenario, and in fact it's one of the reasons I didn't like helping users with their lost passwords anymore, because it's hard to prove they aren't just giving you vague details in hope you can help them get into it. If you post on social media or this forum about someone losing the wallet, you're going to get people claiming to have lost it, and some of them might be able to decipher certain information if you've included it in the explanation. The best way of doing it would be to be as vague as possible,
hey guys, found a wallet this weekend let me know if you've recently lost a wallet, and from there you can kind of filter out the users that are trying a shot in the dark.
However, it's still a massively complex issue, and ultimately the decision you make could be a wrong one without adequate proof. There's not a whole lot of adequate proof if they haven't got backups, and at that point they should just move the Bitcoin to another secure address.
Honestly, I think I'd hate to be in a situation where I find a wallet. I've handed physical fiat wallets, credit cards, and things like this into the police a few times or sometimes just the establishment I'm at. For example, a restaurant because the person might come back to claim it. However, that's easy to verify. Bitcoin isn't exactly easy.