But I just can't think of any fun and meaningful games if you have to design the game so everyone has the power of a cheater.
WoW and all later high-profile MMO projects (indie and similar tend to be a mixed bag).
Zynga, Playfish, Facebook and other social gaming (trivial gameplay which automation devastates).
Also, Starcraft 2 (competitive), Diablo 3 (includes real money, so kinda a big deal), etc...
Whether the games above are fun is subjective, but all of them assume, in their design, the client is untrusted.
FPS in mean time require minimal amount of trust. Since as article above points out that isn't possible, they usually separate servers on those that add third-party checks or other methods to control the behavior.
They could make the game just a flat map with nothing blocking the view except the players.
Yes, that can be done without even touching the game, just intercept the DX/OGL calls. That's a fact.
Yet online FPS genre continues to exist despite that. So apparently it's not that big a deal. You may however require administered servers and additional tools to determine who actually does that. It's also possible to analyze movement patterns to notice who "knows" too much.
All of these methods are more expensive.
Adventure games were a genre that required such secrecy. When internet happened, the genre died out - all puzzles were just a search away. It is quite possible that your idea is obsolete as well. It happens.