How does blizzard achieve it? Well, people can't communicate to the opposing faction and people find groups for themselves. Groups that share their vision on what online communication looks like.
......Really? That's the conclusion you draw from World of Warcraft - that their game is more lenient and less involved with toxicity because players from Alliance and Horde can't communicate with each other?
If you're going to use the absence of cross-faction chat as your point, why bring up Guilds and Raid groups - which, by the way - are faction-exclusive as well. You can't play an Alliance character in a Horde Guild, and vice-versa.
Rather than following this weird non-sequitur, let's take a look at the actual differences between World of Warcraft and League of Legends.
First and foremost: World of Warcraft can be played completely solo, with absolutely no direct interaction with other players whatsoever. You could play from a fledgling, level 1 Human Warrior from Goldshire all the way up into Battle for Azeroth content without having to work with another player. It's completely optional.
Meanwhile, in League of Legends, It's a team game. You can't really play solo, unless you like playing Customs which afford you nothing. When you queue up for a match of League, you're not queuing up for a single-player experience. You're queuing up for a team experience. That's the deal. Teamplay isn't optional, it's mandatory.
Second; Because World of Warcraft can be played entirely solo, it can afford to give players tools to not deal with other players - regardless of potential abuse cases. Running in PUGs? You can votekick someone. Running a group? You can kick anyone at will. Running a guild or holding an officer rank in one? You can remove players that you find problematic. Players being general assholes to you? You can mute them and report them.
League, on the other hand, can only afford the ability to mute and report players. Being given any ability to just boot someone out of a match would ruin the game, since that would ultimately result in it being a 4v5, or worse. Additionally, they can't offer the ability to just kick players at-will, since nobody in the group is assigned as a leader, unless they're in a premade. And even then, they can only boot someone if they haven't queued up yet.
And as for cross-faction play...Like teamplay, cross-faction interactions are completely optional. If you're an Alliance player wanting to deal with Horde players, you can just turn on Warmode and bask in the paranoia. If you don't - you can just leave Warmode off and play PVE. Battlegrounds and Arenas are optional as well, and there are very, very, very few zones that can automatically turn on PVP, all of which are pretty clearly telegraphed when you come near.
In League, a case could be made about removing the ability to chat with enemy players...But that doesn't solve any problems with toxicity. Teammates can still be toxic, no matter how much or how little you can chat with the enemy. Hence, why I call your point of "no cross-faction chat" a non-sequitur.
At the end of the day, though, League of Legends and World of Warcraft are two different games, with completely different gameplay paradigms - League of Legends is a MOBA, which is primarily designed for PVP, and World of Warcraft is an MMO, which is primarily designed for PVE and cooperative play, but can have PVP elements.
If you're going to try to compare games, at least acknowledge the key differences between those games before you try making points based on how you think one works better than the other.
So if we were to unban all toxic players, how do we make gaming environment comfortable for both non-toxic and toxic players* without breaking soloqueue?
"If".
Of course, the prisoners isl...nope, we are not doing it.
Ban their matchmaking.
Just do it. Let them play only in 5-people premades, whatever the queue it is. Ban their allchat. Everyone is happy.
Banning /all chat, or removing it entirely, etc., would not help. As mentioned earlier, you ignored the key differences between WoW and LoL and decided to pick out an absurd, irrelevant feature to make a point from.
Removing a toxic player's ability to be toxic to the enemy team does not remove their ability to be toxic. It only narrows it down. So you've still got people being toxic to their team. Now what? D'you then decide to ban them from using chat altogether? You've doubtless browsed the PB boards long enough to know what comes next.
As for forcing players to either queue up as five or not at all...Why make such absurd restrictions just to try and mitigate toxicity when it really doesn't belong in League? If someone's not able to play on their account because they can never get a premade of 5, they're either going to quit out of frustration, or start a fresh account to circumvent the queue ban. And in both cases, you could've just permabanned the account and achieved literally the same effect.
No stupid posts about spending 5000 bucks, no pointless cruelty.
If someone spent 5000~ dollars and then ran their account into the ground, that's their own damn fault.
As for "pointless cruelty"...I'm pretty sure "reiterating that the rules exist and have consequences for breaking them" and "trying to get a player to stop being an asshole" and "getting rid of players who don't want to learn to play by simple rules" are all very valid points, and a permaban from a videogame is far from cruel.