They could verify those things on other clients, instead of on the server. I think that's typical for pure P2P multiplayer games. They could use something like - every player sending their version of 'what happened' to the server which then decides the proper outcome.

Btw, is it required to simulate everything on the server side? I know it is the conventional way to prevent cheating, but how about just randomly sampling some of the actions to detect cheating, and ban that user?