Well, you explicitly have to clear every bit of (non-Savage, non-Ultimate, non-Unreal) content in the game to have access to Mentor Roulette. Because Mentor Roulette is designed to match you into every bit of (non-Savage, non-Ultimate, non-Unreal) content in the game.
So... kinda, yeah?
More specifically and seriously, it's not that you need to clear everything synced, per se, but that you need to be prepared to run it synced. Which means you need to be familiar with the fundamentals of mechanics (i.e., the LEGO pieces this game uses when building mechanics), and able to read a quick summary of a fight and act on it.
The point of all roulettes is to make sure that people who need to do content have other people available to do that content with. Without Duty Finder, you'd have to go set up a party finder (or ask on your FC chat, or whatever else) every time you wanted to do the next MSQ dungeon or trial as you went through story. We have roulettes for alliance raids, and if you queue into alliance raid roulette you should be prepared to get any of the alliance raids you've done; queuing in expecting Crystal Tower raids may be a safe bet, statistically, but you should be ready to end up in Puppets' Bunker if you're on a level 80 job. We have roulettes for leveling, and if you're on a level 80 job you should be equally ready to be matched into Cutter's Cry as into Mt. Gulg.
Mentor Roulette is explicitly the catch-all, the "this duty needs another warm body to function, but there aren't any explicitly queued in an appropriate roulette" fallback. It is mentor roulette because, since it can match you into literally anything, you should be prepared to be matched into literally anything -- including content you might need to help folks figure out how to clear.
Now, whether or not that's a roulette which should be tied to the mentor system -- and whether or not the mentor system is inherently broken and you have strong opinions on the sort of thing which would be a better system (as I do) -- is an entirely different topic.
But queuing into Mentor Roulette is queuing in knowing that you could be put into literally anything in the game that isn't savage, ultimate, or unreal. Which means you should be prepared to be put into literally anything in the game that isn't savage, ultimate, or unreal.
Yeah, those extremes are all optional... but so is mentor roulette.