Though I've noticed Japan's got a thing for making superheroes into employees or "Public Servants" with ranking systems
I think that the ranking systems come from D&D, or at least Japan's interpretation of it. The original Dragon Quest and Final Fantasy games in 1986 and 1987 were based off of D&D filtered through Japan's lens (and video games), and those initial DQ and FF games codified the "fantasy genre" in Japan (just how LotR and D&D codified "fantasy" in the West). The foundations that DQ and FF laid down regarding the setting of a Japanese fantasy setting (monsters roaming outside town, adventurer's guilds, crystals, etc) were ingrained in the minds of people and are accepted without hesitation. Ranking systems were added to rank monsters and adventurers. I think when One Punch Man and MHA were being created, the authors probably just took the ranking systems from Japanese fantasy and applied them to their worlds without much thought (admittedly the logic makes sense at a first glance). Superheroes in OPM and MHA are pretty much just adventurers from Japanese fantasy, except with more bureaucracy typical of a modern 21st century 1st world nation.
You're supposed to hold back
Japanese anime and games can be weird like that. "No, you can't kill the bad guy, or you will be just like him!"... even though said bad guy is actively trying to kill innocent people and just killing the bad guy is the simplest way to stop him (Xenoblade 1 comes to mind, Tomino Gundam). Or "whatever legitimate grievances you have are irrelevant. Stop fighting!" (Miyazaki's Ghibli films, Pokemon, Tomino Gundam, etc). Attacking people, or god forbid, killing bad guys, is a heinous act that good guys never do. (I mean, not killing people is preferable yes, but often Japanese fiction I've seen seems to take it too far to the point of ridiculousness).
I think BNHA's point regarding "vigilantes" was that they were likely untrained or did not have the same sort of political connections as the ones who now work with/for the government. And that well-intentioned heroics could have disastrous accidents like that one character called Gentle Criminal, who I think accidentally hurt a guy he was trying to save even more.
...
leave it to trained and affirmed professionals more than anything
I always found the outlawing of the use of Quirks in public unless you have a hero license to be ridiculous. When 80% of the global population has superpowers, you can't really stop people from using their powers. You cannot possibly enforce that. Sure, you can arrest a few people, but quirk usage is going to be widespread and you'll never be able to clamp down on that. Furthermore, many quirks are useful.
I also found the idea that "you should never try to save people, even if you are right there, unless you have a hero license!" to be dumb. The example they used with Gentle cannot be taken to be representative of the outcome of these incidents at large (also the example was rather dumb because Gentle had absolutely no way of knowing that a flying hero was about to swoop in from offscreen. As far as he knew, that guy who was falling was going to die and Gentle may very well have been the only person with a chance to stop it. Gentle was absolutely 100% correct to try to save that guy, and the flying hero coming out of nowhere was unfortunate, but not Gentle's fault.
TBF, the heroes of BNHA and series like Tiger&Bunny and Ratman are mostly good people, but that seems mainly because aside from wanting cash, they're fans of what they saw as legit heroes and it helped that those who say, met All-Might know him to be a legit dude. Pedestals intact. Plus, unlike the above, their pay checks are probably slashed the moment they cause trouble.
MHA's worship of the state always felt immoral to me. In season 4, a little girl runs up to Deku and begins crying for help, and an obvious bad guy walks up to Deku and asked him to hand the girl over to him. Alarm bells are ringing in Deku's head (rightfully so) and he's contemplating fleeing with the girl (and then sorting out if she really belonged to that guy later). But then the fellow hero who is next to Deku is watching this all happen and is doing nothing, and crapping on Deku in his head because legally Deku isn't supposed to do anything. The show treats the right thing to do is to just let the obvious bad guy waltz off with the little girl. It's bizzare. (Now yes, they were staking out the bad guy, and he was pretty powerful, but the narrative seems to condemn morally right actions if they break the law).
That said, my beef with the whole fame and celebrity thing, is more that they make it a lot more official(by working with corporations and doing ads)and make it that most heroes are in on the whole getting sponsored thing.
Yeah, that's another thing I'm not a fan of. MHA seems to have this circus about heroes trying to become popular and celebrities and chasing stardom, and the narrative seems to treat this as if it's normal, like that this isn't a vain or selfish thing. In S4, they have the 10 top heroes get on stage on national TV and it's treated like "these are the protectors of our nation!" when it's really just a big popularity contest.
There is also something nihilistic about MHA, where one of the top 10 heroes, Hawks, decides to try to spin lies to the public and try to manipulate the perception of Endeavor, trying to promote him as this big, moral, mighty protector like All Might as "the hero of Japan". I mean, Endeavor isn't All Might and lacks charisma, but the narrative doesn't seem to believe that such people can become admirable community figures without the aid of liars and subterfuge.