The hats in Find Mii/FM2 are different than the hats you get via Plaza Tickets.
You don't have to play any of the games to let more Miis in, you get a message saying there are new Miis at the gate, but if you greet them, you lose the ability to "play" with the previous Miis.
Note: There's still a limit of ten StreetPasses before you have to greet them to allow more to come to your gate.