Do all (or most) gamecams have the tripod insert in the bottom or just that one?
That's a tough question since there are so many cameras out on the market. I would say MOST probably DON'T have the insert in the bottom, but look before you buy one as several brands do have them.
You could also rig up a plate that would allow you to mount a camera to it and then screw the plate into the bracket pretty easy if you have other cameras already that don't have the adapter hole in the bottom.