My usual suggestion for that is to score people based on their recent match history for matchmaking. The short version, if you play one match as tank, you get a pass for an assassin game. (Further explanation below)
Ever since my main mode is QM, I even stopped calling it Slave League for forcing me to play tank or healer too much.
I suspect that many people are like me actually, perfectly fine playing some games filling all roles, just not long streaks of it.
I don’t know specifics but Grubby mentioned a role queue in DotA2 which is similar, you can queue for a specific role every other match or so.
I see a lot of people enjoying just a subset of heroes (but way fewer OTPs these days, possibly because I gained rank or I’m flexing myself), arguably ranked isn’t for them, yet they might play with a friend or simply enjoy the ranking process. (Same for me, but Heroes Profile does it.)
The explanation. I don’t know how simple SL matchmaking happens to be, but I suppose it does the same things as QM, with some parameters not active of course. It certainly uses rating and waiting time, plus rating mirroring (when parties bring bronze + gold, both teams tend to have this, ideally party sizes as well).
I think what it does, it tries to score match candidates somehow, starting the best match or combination of matches.
So the trick is exceptionally simple, with some deep details. In generic version, if your role pick cooperates with the team, you automatically carry some MM score with you, meaning you are likelier to be included in a match sooner. A fairly transparent version is to automatically bump wait time value by 5-10 minutes, so you wait that much less.
I mentioned cooperation because the real annoyance is “first pick Ming”. Sometimes I do get teammates showing tank-bruiser-healer from the get go. In this scenario, picking Tassadar / Ming is filling, even as first pick if the others did show.
Anyway, that was just a bit of a brainstorm about how I’d tackle this. Just in case, or for my own fun. Also feel free to attack it, I’d benefit from it professionally.