I'm trying to play with a friend. However, after I change the entry permission to "public", the game automatically starts in 60 seconds regardless on only two players being in the lobby. We can't try to get into Quickmatches together, their are rarely open slots to invite one another. Is there no way to control a Public lobby?

Public lobbies don't allow for any control, unfortunately, which can be a pain if you don’t have enough friends for a good match. I'll admit I haven't had that many issues joining friends in public lobbies using the online menu, but I'm sure that's not much solace for your situation.

Criterion's games seem to have finicky public lobbies in regards to filtering players in. It's unfortunate.