I think trapping has changed a lot since Team Preview was added in BW. I didn't really play in that metagame, but as someone who runs bulky teams that often have a potential Goth victim, it's nice to know that it's there. Personally, I think it adds a fun layer to the game, and the predictions and double switches can get pretty zany. Of course, I'm not a stall player, so it's a lot easier for me to enjoy that aspect of the game. If Goth can trap and kill four of your Pokemon and requires very little skill to effectively use, I can't blame you if you're annoyed with its presence.
I find it a lot more annoying in DP and especially in Advance (not like I've played that since a half decade ago, but whatever), where the very prospect of Dugtrio and Mag made you have to play differently. And in DP, with the rise of strong Pursuit trappers like Weavile (bad overall, but difficult to deny its trapping prowess), Tyranitar, and Scizor, that uncertainty was extended to a lot more Pokemon as Dug usage fell off. I don't like the idea that something can trap you without you knowing it's there. Maybe this belongs in a thread about Team Preview, but I think that it makes trappers more fun to play against when you know what you're up against. If your Latios gets Pursuited by Aeg, tough, but you knew it was there and could be carrying Pursuit. the only advanced warning you have pre-BW was metagame knowledge, which isn't exactly the same thing.
Anyway, regardless of how fun or dreadful they are to play against, I think they're pretty clearly uncompetitive, and I think D_A did a good job outlining why. I think the distinction between uncompetitive and broken is an important one. Double Team probably isn't broken (?), but it's definitely uncompetitive. I'd slot the inferior trappers (even PROBOPASS!!!!!!) there. I do agree that Goth is broken.
Besides the choice limitation that D_A talked about, I think Pokemon like Goth encourage the "hey, XY OU is way too team match-up reliant" reality that seems to get worse with each successive generation (and was probably exacerbated by the glut of borderline-broken but at the very least very threatening megas) that I think is somewhat natural, but not to this extent. Goth is just horrible against offense, but can take apart stall with anything competent to clean up its mess. A lot of Pokemon are significantly better vs. one end of the offense-defense spectrum than the other (both genies come to mind), but they also don't trap things. You can run niche stuff on stall to take out Landorus or at least lessen its impact. You can run Ground-type Scarfers like Excadrill and Garchomp to check Thundurus. But since Goth is by definition uncounterable, I think it's kind of a cop-out. Stallbreaking ability is a decent barometer of competitive skill, and Goth is an easy way out to break up strong defensive cores, just like BP was an easy way out against...well, most everything. I don't think it's as bad--Goth can only do so much and isn't near as uncompetitive or as broken as BP was (DING DONG THE WICKED WITCH IS DEAD), but I'd like to see it banned. I don't think it's a pressing problem like BP was, but again...I'm not a stall player. :)
As for past gens, DragMag was absurd and I do think Dugtrio is pretty uncompetitive in Advance. If the way teams are combatting trapping is Trace counter-trapping with Porygon2, then it might be a little much. Of course, as with the Mag/Skarm relationship Moxie discussed, trappers can keep difficult Pokemon in check (Dugtrio with Celebi/Tyranitar also comes to mind), but I'm not sure that's the best way for a metagame to balance itself. I can't speak with much authority on older gens though because I'm so far removed from them.
#FREEPROBOPASS
Also D_A, not going to disagree with most of what you're saying, but didn't you say in early XY that Mega Gengar wasn't broken? EXPLAIN YOSELF
Why are the beautiful sick and divided like myself?