I have to disagree with some of those.
Trundle is an exceptional duelist, his ultimate/q go a LONG way in turning the tides of a fight, and his passive is constantly activating for significant chunks of health as battles go down. His mobility and his pillar are also very very useful.
Tankwick can also win games. He's very difficult to take down, his Q regens a lot of life and it makes him invaluable for tanking towers to take them back, and defending towers from aggressors. He can also see into the jungle if someone is at low health from nearly across the map, which is also handy.
Teemo is ok. Sweeper Teemo butt****s stealth champs like no tomorrow and his blind is as infuriating as ever, but otherwise he isn't that special. A low priority pick at best, but not completely worthless. Also, I'd rather have Ashe. Way stronger poking, isn't melee, and dat ulti.
I agree on Shen. He's just weak overall right now though, some buffs will make him completely viable in Dominion in the future. But for now, he's too overnerfed of a champ.
Ashe is definitely viable. Her ultimate is extremely good on Dom, can kill people she isn't anywhere near if they are trying to cap a turret. Not to mention it's great for initiating onto points safely. Hawkshot is also one of the very few abilities that can see into the jungle, so that is very worthwhile as well. Volley is a great poking mechanism and perma-slow is pretty strong on Dominion as well. Best pick ever? No, she definitely has her weaknesses. But she is by no means bad.
Veigar I can kinda agree on... he can straight up one shot people with his ult up from a very early point in the game, but once it's down his power falls considerably. He can't farm up his Q, but the trade-off is that he can't be denied any items and scales very powerfully with the items he gets. Again, he has weaknesses, but he's not worthless either.
Garen is good on Dominion. Really I don't see how that's entirely debatable, Garen is an exceptional duelist and his silence/spin is the bane of casters. He may not farm his W all the way up but then again, he's not so heavily reliant on that that it breaks his character. He does not scale as well into late game, but by that same token, he's a fast, hard to kill bruiser with an aoe death spin and an awesome execute that rapes hard early game. I don't see how he's bad.
Lux also is not bad. Kassadin does rape her, but what caster does Kassadin not rape? Kassadin has been a top ban in SR/Dom for ages now for a very good reason, he just makes a caster's life far too difficult. She can be screwed by gap closers, but so are so many other champs.
How is Corki bad? Corki in my eyes is one of the stronger ad carries you can bring into the fight, he has significant damage and his poking is incredible. He has decent mobility and nice armor shredding, I don't see how he's bad either.
I'd go ahead and say Mundo is rather weak. AD Mundo can do some work, but I'd much rather have someone like Riven or Olaf who are way more reliable, and have so much more going for them. The heal reduction aura hurts a lot too.
Nunu can be decent, but I think he isn't especially amazing either. He is very scary when paired with someone like Vayne or Yi, but at the same time idk. There are other supports that are just as scary that do even more like Janna or Sona.