Not to mention that they would find a way. You see, the game data would have to be transferred to the computer for game to run. While normally it would only be cached and cache wiped after playing to prevent piracy as you say, they would find a way to stop the wipe and run it off the cached stuff.
Now, if they actually RAN whole game on servers and streamed it to you, then it could work. but... that would require few things: lot faster (both lower ping and higher bandwidth) internet connection and MASSIVE increase in hardware on servers. right now they wont need to do graphics stuff, If streaming, they would.
Pretty much every player would need to have their own "computer" even if virtual on the server.
So.. no. as far as ideas go, not a bad one but it simply wont stop piracy only hinders the normal customers who bought the game.