Before I get to any of the meat here, let me pick the first sentence right off.
punishment teaching systems are a broken concept of teaching and has been proven to not work.
The punishment system isn't meant to teach. It's meant to punish players for breaking the rules and show them what they did so they themselves can readjust. And, more than that, the punishment system is meant to punish players for misbehaving - and since the audience is roughly teens to adults, composure and sportsmanship should generally be intrinsic traits.
Of course, the facade of anonymity brings out the absolute worst in some of us, but nevertheless, the punishment system isn't there to "teach" people how to be sportsmanlike and display common decency. It's there to reprimand those who can't and don't.
riot needs to help teach and guide the players to what is a good honorable player than just banning them progressing the problem.
I kind of (and I do stress kind of) agree with the first half, but disagree with the second half. While it could help for Riot to have resources to show sportsmanlike and honorable behavior, having such resources should not allow dishonorable players a free pass to break the rules.
Bans only "progress" the problem in the worst of cases. Most folks who get punished never get punished again, and those who do a second time seldom do afterwards - and the group size decreases with the number of punishments. There have been cases of people who (at whatever stage of the punishment ladder) decided to throw everything out the window and be toxic simply to spite Riot or the community that slighted them, but they're more or less outlier cases or worst-possible extremes.
I think a good thing to try is a class like thing where the players are taught what how to be a good player through games and situations and the whole time they are being tested and when they are shown to be good enough to pass the class they can play league again.
Again, first part - a very, very tenuous maybe. I don't wholly agree with there needing to be a class on honorable/sportsmanlike behavior, because, as I said - if you're playing League of Legends, odds are high that you're old enough to have been taught it already. It should be intrinsic, it should be natural to be honorable. You shouldn't have to take classes on it.
The latter part, though - about testing them and only allowing them to play League afterwards - that's extremely arbitrary, and potentially a worse punishment than the actual chat restrictions and account suspensions doled out regularly. It's kind of like an indefinite temp ban, with an extra layer saying "here's a course on sportsmanship and honor! pass this and you can play again!"
And, by likening it to a temp ban, I mean that players who suffer such a punishment are extremely likely to sidestep it by creating an alt and playing from there, or just quit outright. It potentially loses players who otherwise would have easily reformed.
players that return to this class more than once show that they are not going to change and can be punished for being a bad influence to the community.
The system we have already does that. If you get punished multiple times, your consecutive punishments escalate in severity (10-game chat-restriction, 25-game chat-restriction, 14-day ban, permaban) because you are regarded as an increasingly bad influence to the community.
The community is what shapes the game and the community is shown by popular streamers that bming is OK and killing someone and using and spamming an emote this shows that being a player that makes fun of other players is an OK thing to do teaching everyone that its fun to ruin the game for the opponents.
I for one don't watch streamers, so I'm not quite so up-to-date on who's the newest toxic D-Bag, but I do know some hearsay on that matter.
Further, there's no rules against competitive BM as long as it's within reason. Flashing an emote every time you kill someone? A-O-K. A bitter jibe or two at the enemy you're dominating? Keep it brief and don't go too hard on them, and you're clear.
Making fun of other players, though, is against the rules. However, if streamers are making fun of players verbally (I.E., not in League chat), there isn't really anything that can be done about that.
And, lastly, is it not the whole point of winning to, in some small way, ruin the game for the opponent? Like, sure, singling out, crippling, and decimating a single given opponent is pretty far over the line, but if your goal is to win, you're probably disinclined to give two shits and a rat's ass about your opponent's game.
Altogether, I don't see anything like your proposal happening anytime soon - especially not as a replacement of the existing punishment system. Maybe some day in the future Riot will make resources for players to better understand honorable conduct, and I imagine sometime soon they'll make tweaks to the "reform" part of the punishment system, but, as it stands, what we've got is the best we've got.