A lot of it is a generational thing, IMO. Wolverine in the Byrne/Claremont/Austin era and the Frank Miller miniseries >>>>>>>>>
He was great back when he was written by the people that knew what they were doing. (To be fair, most of the X-men got pretty badly messed over by subpar writing at some point, especially Scott with the editorial mandates surrounding early X-factor)
The problem is when all the stans who grew up in love with Wolverine as kids got old enough to write comics. They didn't realize what made the character work. They made him a boring invincible hero who could regenerate from atoms (Piccolo status). They demystified and watered down the character by shoehorning him into every other marvel comic out there, and overusing him in the movies, cartoons, etc. Telling tons of stupid stories that don't need to be told about his past when he's really the Raphael of this whole X-men business.
The people that understood the character got that he worked initially because he was an enigma. He was a walking noodle incident, you weren't supposed to know everything about him.
Wolverine's healing factor had limits, his temper got his teammates in trouble as often as it helped them, and they wouldn't even show you what happened to his opponents half the time, but you still KNEW they got murked. He was a cosmic-level trash-talker but got served up with the quickness whenever he stepped out of bounds.
He functions best as a side character who occasionally gets his shine in the limelight.
Making him the leader makes no sense because we all like Logan for being brash, crude, rude, a lone wolf, and unwilling to play by the rules.
You can't be those things when you're responsible for everybody's lives.
EDIT: Example-- THIS is Wolverine. They don't show you exactly what he does in either scene, but you know those dudes got murked. It's like in the movies when you're actually MORE scared when they don't actually give you a good look at the monster that's chasing everybody. Your mind has a way of filling in the blanks with way nastier stuff.
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So when you finally see him ACTUALLY catching bodies at the end of the Dark Phoenix Saga, you know things are serious.