Let's check out them immobile melees and their CC:
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This is a CC guy, that CC is usually better spent on peel rather than engage though.
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This is THE CC guy, he wants to teamfight as often as possible. 4-man ruptures win games.
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This guy got a bit shafted, yeah, he was transformed into a one-trick pony, where his one trick is winning lane against a melee champion, even then, if the enemy carry is sorta dumb, you can pull him into an early death, or if you're ahead of your backline, you can pull an assassin from your backline to the frontline, peeling for them.
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A CC guy, a teamfight guy, he wants to five 5 people and ult, teamfighter.
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This guy got shafted as well (Even moreso with the removal of DFG), he's a champion who needs help desperately, since he doesn't have a clear niche (a CC-less tank?, an immobile bruiser).
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I don't know HOW immobile he is, since he gets a dash and an MS boost, but he seems pretty kiteable after using his ult, anyways, this is a guy with a lot of CC to do teamfights.
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As I said before, it sucks to be a melee mage with no mobility in this meta, he needs help.
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This is the one immobile melee who should be splitpushing, mainly because he's hard to kill, usually carries ghost as an escape, can nom-nom turrets in a couple hits and is generally impossible to duel come late-game. Even then, with enough CC on your team, he works great as a carry buster during a teamfight, since if the enemy team is wasting it's CC and damage on a Nasus, your team is winning.
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This is a guy with a lot of slows during teamfights, and a nifty buff for your ADC, so more of a teamfighter, yes.
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The guy got Olaf'd, so he's not as useful all around as he used to, even then, he's a guy that can turn red, run for the carry, and guarantee that the whole enemy team is trying to nuke him ASAP (no chain-cc here). If your team can capitalize on that, that'd be great.
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This guy is technically an immobile melee, lot's of CC, great teamfighting, he's cool.
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Lot's of CC, interesting teamfights, if you MANAGE, to snag any priority target, that's a dead target, I have also seen his ult used as peel against a Zed bursting your backline, it's beautiful.
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I guess he has a bit of CC?, he should be by your carry, peeling for him, healing him, and boosting his stats. teamfighter.
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Admittedly, this guy works better as a splitpusher than a teamfighter, cool duels, viable escapes.
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This champion is overall terrible at splitpushing, since he pops turrets very slowly and without fantastic waveclear, while having no relevant escapes, his teamfights, while lacking in CC, have a lot of utility with that ult.
So indeed, most immobile melees actually DO have access to CC, and work better as teamfighters. One reason we won't be seeing them that often in competitive play, is because teamfights aren't all that common in the competitive scene, this does not mean they are useless in non-competitive play.