A no press game is a game with no press. :P That is, all communication is cut between all players.... But that's what not this game is about, and, sorry to say, we're already full.
Normally, yeah, I agree that it would take longer for a WTA, but, since everyone will be consciously making an effort to speed things up, I don't think WTA will be a problem, and WTA would make for a better game, I think.