I think thats a little bold to say mentally unstable act like criminals. Maybe the most extreme id be willing to say that about, but theres mild cases where people arent at all unlike you and me but still would be harmful to themselves or others if given a gun (suicidal, bipolars, etc). again this seems to be the biggest disapproval of any such screening process, "itll just take me longer to get my gun," which is true but really just comes down to the owner not wanting to be inconvenienced vs a system that would actually be beneficial to the whole. its not "gun control" limiting you from getting your gun or even saying what you can get, itll just take you more than a day or two to actually get it, which i dont think is asking that much.
Mentally unstable folks will act very much like criminals. They wont care about the legal way to obtain something and, instead, will opt for whatever means of getting a gun to accomplish their goal. The only thing such a meeting will do is slow down the purchase for already law-abiding, sane citizens.