I'm afraid that my two cents is going to have to diverge from the pack. I'm not going with a card-drawing card like copycat, oak, or elm. If you asked me "What would you do without copycat?", and the format were unlimited, I'd say "I'll play oak and elm". If you say, "What would you do without oak?" I'd say "I'll play copycat and elm", etc. I certainly don't deny the power of card-drawing; I run just as many of them in my decks as anyone else at a competitive level. They are vital. However, the question of whether they will change the course of the game is answered solely through chance. They will improve your odds of winning, but I've played enough games to have been soundly beaten by people who never once use a single card-drawing or searching card (ala CPU or IF), and I've lost plenty of games in which I was in need of a certain card - most often the one I'm going to nominate in just a bit - and used four or five card drawing cards (Copycat...darn....oak...darn...oak...darn....cpu search...oh, there it is...only took me sixteen cards and half of my draw engine to find it) just to find it, if I find it at all. My nominee for best trainer is the one I think changes the game the most profoundly, and that is:
Gust of wind.
and, in modified, its younger brother double gust.
With the advent (birth?) of the baby pokemon, this board-altering card has become even more important, especially in tournament play. In the case of GOW, there are no drawbacks, there is no chance of a result you do not intend, and no flip. Even the low-drawback bill has a chance of yielding an early game nightly garbage run (town volunteers, if you prefer) and basic energy. You simply change the board in a way of your choosing which favors you. It will get you the knockout. It will save you for a few turns. It will shut off active pokemon powers. It will force your opponent to burn cards to counteract it. To close, I've asked this question near the end of games hundreds of times: "Do you have the gust?". If the answer is yes, the game is over. I can't recall a time when I've asked, "Do you have the copycat?" and crossed my fingers hoping the answer is "no".
Addendum: I can see someone arguing "Yeah, but how're you gonna get that gust in your hand, huh?" My reply: any one of a dozen or more ways : oak, bill, copycat, POR, comp. search, item finder, etc. I retort: "How are you going to get that benched pokemon active for your final KO this turn?" Unless you're playing Victreebel/dark ivysaur, or you like flips, or you're willing to wait a turn, you have two options, both containing the word "gust".