How do I play video through a projector without seeing the player buttons?

Both Quicktime Player & VLC will show the cursor & transport if you move the mouse, but both will respond to a simple Spacebar tap to play/pause without showing the interface. VLC has an optional || or > indicator.

If you never need to use playlists or run consecutive movies, then stick with Quicktime Player, otherwise, VLC is a much more comprehensive structure.