Why the heck would you use only Weedle? Maybe Caterpie since Butterfree is pretty good, but Beedrill couldn't even beat Brock even if it was level 30. You can do it if you want to, since I know some other speedrunners who do their own unofficial categories (i.e. Speedrunning Pokemon Emerald: Complete the Battle Frontier,) but I don't think that should be a category. Do it if you want to though.
P.S. Besides, don't you use a starter Pokemon in the beginning, so it is not only Weedle.
anything can become a category if enough people run it and it's somewhat competitive
That is somewhat true. Most types of a speedrun can be an official category if people want to do it. A category with only one Pokemon would be interesting to see, but I don't think most of them would be possible. For example, if someone is doing Weedle %, (beat the game with only weedle,) won't you have to use a starter Pokemon, where one of them isn't a weedle? You would have to cheat a starter to be weedle. It could be done, but I don't know how many people would want to run with only Weedle, or Jigglypuff, or MissingNo or whatever. One more thing: Just because there is no Weedle% category or Deoxys% category, it doesn't mean you can't attempt it. Lots of speedrunners do speedruns for their own insane category.