It might to some extent, but the same would be true even if you changed the trophies. Chances are the trophies wouldn't different drastically league to league so the players who go for Dragon are likely doing it for more than just the trophy. Even if they don't get a separate item in the main game, they'd likely still be interested in going for the top ranks and they would be the type to appreciate more text in a read option on the trophy. The players who would just get Dragon once and stop caring will likely do that regardless of how many variants of trophies you have.
Also, another issue to bring up is how cluttered the game could get if every league does have unique trophies. Five leagues in, they'd either all look rather similar or all try to be unique, which would be worse when you consider how many are now unaccessible to all new players.
The last thing I didn't think to mention in my first post that I'll tack on here is that it really isn't fair to change how the prizes work halfway into the competition. Players went into the Twisted League being told nothing would be unobtainable in the future, so if a month in it is changed so the Trophies are unobtainable, that is a bit of a slap in the face since anyone who would change their priorities based on that is now a month behind. For example, I've been playing leagues fairly casually, but if I went in knowing the trophy was unique, I'd have 1000s of more points by now because I would have put in more time.