Part of me thinks everyone should have to push the button to skip in order for a cutscene to be skipped.

That's how Monster Hunter 3 Ultimate handled it (Though, that game only has a couple cutscenes in multiplayer)- in order to skip a cutscene, everyone in the group has to press the skip button. This way, if everyone is skipping it you can skip it, but you can't continue on without the person watching the scene.