I also believe they're trying to gain control of this site, or at the least set themselves up as THE lyrics site.

If they were protecting their clients' rights, they would simply have given Max the information he requested - what material they found objectionable. They say they will do this when they file charges. Why do they want it to go that far when he's perfectly willing to work to resolve the issue? Possible reasons:

They don't know what songs are in the database because they've never been there. They quite possibly have not had any specific complaint, and are going on an assumption that if it's a lyrics site, some of their clients' songs must be included. This would be a reason to ask Max for a list of songs.

Their main goal is NOT to get the offending material removed, thereby protecting their clients' rights to have their songs made available to the public for free only via the HF website.