Lol I'm late to the party :$
Swirlix: Broken
Unpredictability: This is pretty much the main argument for anyone wanting to ban Swirlix, and that's because thanks to Swirlix's role as a late-game sweeper, being unpredictable is a great asset to have. Until Swirlix enters the game, the one facing it has no idea, firstly whether it is physical or special. This leads to the one against Swirlix having to play a guessing game on which of his Swirlix counters to try to keep healthy, and guessing wrong means you lose. Secondly, you have to hope Swirlix isn't carrying the fourth move that allows it to beat your counter/check, or that if you see Calm Mind and plan on checking with something like Meditite, it doesn't carry Cotton Guard as well. Think of Mega Lucario in OU, where there's a sweeping set on both sides of the spectrum that can sweep you with ease if you screwup one time.
Ability to Defeat Counters and Checks: Swirlix's ridiculously wide movepool allows it to beat normal Checks and Counters with slight changes, mainly in the fourth move selection. Common counters to the Calm Mind sets are Croagunk, Slowpoke with Thunder Wave, and whatever poison type you like most. All of these can be beaten with a move Swirlix gets, usually either Psychic or Thunderbolt. For both the Belly Drum set and the CM set, Protect can be used to keep Fake Out users from revenge killing you, and when priority is nearly always the only way to stop him, good luck. Belly Drum also commonly runs Theif to beat things like Slowpoke and Tirtouga that try to counter or check Swirlix. Cotton Guard + CM stop Swirlix's main check, Meditite, from being able to beat it after one Cotton Guard, and it also stop Pawniard. This just adds even more layers to the guesswork involved in keeping Swirlix from sweeping you late game, and it also allows Swirlix to come in mid-game, lure in a counter like Croagunk, then take it out with Psychic to Tirtouga can sweep you late game.
Ease of Set-up: Unlike every other set-up sweeper in LC, bar the imo broken Sturdy Juice Smashers, Swirlix typically finishes you off with full health, thanks to Berry Juice healing any chip damage you did to it when it set-up/healing the Belly Drum, as well as it being at +2 speed, meaning you can only hit it with priority. It also has just enough bulk to be successful in doing this, taking any non SE non LO boosted move with ease. This means there are only a select few Pokemon that can revenge kill it, and like I said before, it can get past these with either Cotton Guard or Protect.
Does Swirlix Make LC Not Fun? Yes, nothing is more annoying than playing a near flawless game, only to guess wrong with what set the Swirlix is running, then proceed to be swept by it. Or playing the whole game to leave it your Meditite against the opponents Swirlix, then have it set-up a Cotton Guard. In addition, you can just have revenged, say a Gligar with Scarf Chou's Ice Beam, then have Swirlix come in, Belly Drum, and sweep you since you can't do anything to touch it.
Does Swirlix Deter People From Playing LC? Yes, because like I said, feeling like you literally can't do anything to beat something makes playing a tier seem stupid, since it makes it feel like skill doesn't play a factor, only who can set-up their Swirlix first, and hopefully the opponent doesn't run the set that can beat me.
Gligar: Broken
Versatility: Whereas Swirlix is just unpredictable, it's pretty much going to be sweeping no matter how you look at it, since Spritzee is a much better Fairy types wall. Gligar, on the other hand, can just do everything. While personally I don't think this makes it broken, it's more the effectiveness of some sets that push it over the edge for me. Gligar can set-up rocks for you to stop Sturdy + BJ abusers. It can Defog for your own abusers of the prior strategy. It can just be a bulky pivot with amazing speed and offensive pressure. It can sweep with SD. It can pass attack boosts, speed boosts, or subs to other sweepers on your team. It can just come in and spam Knock Off for your other sweepers. It can be a wall with Roost and an Eviolite. Literally the only things Gligar can't do is be a special attacker, cleric, or TR sweeper.
Ability to Defeat Counter and Checks: Like Swirlix, Gligar can just screw over your counters to it. It also has a crazy amount of ways to do it. For starters, it can just U-Turn out into a counter. It can also use Knock Off to cripple them for later. It they can't take boosted moves, then SD does the job. The part that really pushes Gligar over the edge though, is the fact that it can SD/Sub/RP on the switch into a counter, then Baton Pass to a Pokemon that can handle said counter, and possibly lead to that Pokemon sweeping.
Stats: Gligar's stats are a giant reason of why it's so threatening. It's bulk lets it take pretty much any neutral physical attack, and it takes special attacks alright too, such as Slowpoke's Scald. It's attack gives it great power even without a boost. The stat that matters the most however, it it's speed stat, letting it speed tie with things like Murkrow and Misdreavus, and meaning many times you'll find yourself relying on a speed tie to beat Gligar, unless you run a Scarf, Elekid, or priority user. Unlike threats such as LO Murkrow, Gligar has decent enough recovery, as well as no recoil damage, meaning you can't just sack things until it can be taken out by priority.
Does Gligar Make LC Not Fun? In my opinion, it didn't. Unless it runs Baton Pass. At that point it's like Swirlix in a way that you feel powerless against it. Thanks to its offensive power, you won't want to leave in whatever you have against it, but if you switch, it could boost then pass, and you can't stop it.
Does Gligar Deter People From Playing LC? Yes, when people see a formerly UU Pokemon in LC, they probably think the tier's going to be horrible, and just a Gligar fest (Which it kinda is actually, but honestly using the reason "Why not use Gligar" as means to ban it is terrible since that can go for lots of Pokemon like Pawniard and Rotom-W in OU.)