There's a difference between being mistake free, and not being an embarrassment to the species. The latter of which I see mentors fail far too often.
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I mean while that dungeon is spicy, someone experienced should have no problem healing it if the tank isn't trolling with their mitigations.
That aside, just because someone's a mentor doesn't mean they're good at the game or their job. Commendations are easy to come by if you play long enough and aren't a measure of skill or 'doing well' because not everyone will judge and commend you based on your performance (or at all).
I do think you should've not kept pulling though when explicitly asked not to. Perhaps they weren't comfortable with the job (trying a new job) or experienced enough to be able to keep you alive. I can't say much about that since I didn't see your gameplay or theirs. Personally, speaking if we wipe to a big pull, I always ask to give it another shot because I sometimes know I mismanaged something or figured I could've used something, I do also understand that people might just want to be comfy and not suffer- I respect that (even if I prefer big pulls and get bored with small pulls).
I've seen many legends eat stacks and warm the floor too. There's alot of content in this game, human error is a thing, even if you're good at specific content. Doesn't mean you'll optimize for every dungeon.
You have to understand that being a mentor is just a mean to get a very nice mount for a lot of people.
That being said, Bardam's mettle is one of those few dungeons that have risky wall to wall pull and you better listen to you healer (or tank for that matter) if he says he can't handle them.
Because mentor only means you played for a long time, and most mentor don't even display their crown. More often than not, one that proudly display the crown is one that farmed commendation with guild heist. You probably have played with plenty of good mentors, but you just don't know it.
Also, you can get mentor without ever touching a heal (smn give you free heal, for example). If your healer tell you he is not good enough to deal with it, mentor or not, you listen to him. Don't play with PU if you only want to play with good player. Honestly, by how you describe the situation, it kinda looks like your mentality is the problem here. You don't seem to even understand why you got voted out.
I've had my crown since Stormblood and haven't worn it because I never wanted to be associated with mentors. But I'm leveling Black Mage right now and it's a struggle. Maybe if I wore it, I'd have an excuse to be doing garbage damage and eating mechanics left and right.
Its always amusing to see people who have a massive axe to grind with mentors.
Mentors are just glorified ingame wiki, not badass hardcore players with all ultimates cleared...i mean, you do not need to clear nothing ever to became a mentor, but just play a lot...thats it