Bronzor and Shroomish etc are not specialized counters to Drilbur. Bronzor also serves as a check to Krow, Dragons, Porygon, and many other Pokemon. Shroomish serves as a check to Chinchou, Staryu, Ferroseed, Tirtouga, etc. Sure there choices that can check individual mons from those lists better than Bronzor or Shroomish can, that doesn't mean that Bronzor and Shroomish usage can be attributed solely to Drilbur's presence. I don't carry any of these "specific" counters to Drilbur, and I still do fine against it as long as I keep up offensive pressure to prevent the opponent from stacking spikes.
Drilbur is hardly unbeatable. Proper team support is never an argument for a Pokemon's tiering. Clamperl is impossible to beat with "proper team support" (removing / weakening checks and priority and providing a setup opportunity), but nobody cries that it's broken. Drilbur needs those same two things provided for it in order to sweep 100% of the time. In fact, as far as setting up a win condition goes, Scraggy requires far less support in order to sweep, and is arguably a better sweeper than Drilbur is.
Drilbur is the face of sand, but I'm pretty sure that Sand teams wouldn't die out if it was banned. They would see a big drop in usage, but Hippopotas is a good mon, and it helps Rock types like Tirtouga set up. So I would say both Drilbur and Sand in general contribute to its success as a playstyle.
I would hardly say Drilbur has a fantastic movepool, It basically has 6 usable moves. They are the moves that it needs to be effective, but if they weren't we wouldn't be having this discussion would we? Mold Breaker Drilbur as an option for a spinner is great for the metagame, and SD Drilbur having 2 options for a coverage move outside of EQ / Rock Slide isn't dangerous enough that we should call it's movepool bad for the metagame.
It's true that Drilbur is one of the mons that cause Mienfoo to run defensive spreads, but even if it were banned Mienfoo would still go a defensive route because of Scraggy, Tirtouga, and other sweepers that it can easily check with defensive investment. A metagame without Drilbur would probably be a little more offensive because it would lower the usage of Bronzor and Grass-types, but it wouldn't knock Foo out of it's #1 spot in usage.
I don't really thing Drilbur warrants a ban :/ There are far more threatening sweepers out there, like Scraggy, Shell Smashers, and Dratini. Even when I don't pack a counter to Drilbur, a shaky check is usually enough unless the opponent gets a load of spikes down, and that is a problem I have with Spikestacking, not Drilbur. I don't think removing it would be a healthy change for the metagame, because the loss of SD Drilbur would only cause a drop in Sand usage, Bronzor, and Grass types, while the loss of Mold Breaker Drilbur as a spinner basically forces teams that want a spinner to use Staryu. I'm not saying that's a bad change overall, but it's hardly necessary.
Twas bryllyg, and ye slythy toves
Did gyre and gymble in ye wabe:
All mimsy were ye borogoves;
And ye mome raths outgrabe.