I'm pretty sure this is because of your subpar mail carrier. I used to work in an office in midtown NYC where we had a great mailman - he did a great job. But whenever he was out sick or on vacation, we would have a substitute, and she didn't know our route like he did. Every single time the substitute was working, we would suddenly start getting all the mail from the previous tenants from several years ago. And it was always a shock to the system to open the mail those days because the previous tenant was a sort of sketchy D level talent representation company that repped washed-up stars like Motown singers that no one has thought about in decades, action stars who were big in the 80s, rappers who went to jail to do serious time, that sort of thing. So all of a sudden we would start getting really bizarre fan mail written by really strange people (you can only imagine the sort of person who still writes fan mail to Steven Seagal).
Anyway, it was clear in that case that this mail-from-previous-tenants thing was something that the mail carrier was dealing with on a daily basis, because it was getting done every day except when the sub was there. Yours isn't willing to do it for whatever reason. Not sure what the best solution is, but I'm pretty sure that's where the problem is.
And yes, we did have people show up on two different occasions looking for the previous tenants, including law enforcement, plus a few times we got various types of certified mail serving legal papers for some kind of lawsuit. Everyone was very understanding, though, I think they're used to people like that being evasive and not updating mailing addresses.