This game has serious free-rider problems and it's getting worse. A suggestion.

[Ronduwil]

This game just has a notable minority of extremely entitled, aggressive karen types. Every game has them, but in FFXIV people actively white-knight for them when they start going off, so they become bolder and bolder

That may be so, and I'm sorry that you've experienced that so often. Personally, I've never seen it, but I haven't done any of the endgame yet, and maybe that's where those folks hang out. However, that has nothing to do with the OP's suggestion, which was to massively overreact as follows:

Suggestion to the devs: expand “FATE-like” performance based rewards to most aspects of the game.

At the end of an instance, you’re rewarded with a medal based on a general measure of your contribution to the party depending on damage, healing (excluding overhealing), damage mitigation and resurrection/raise casts. If you don’t pass a certain threshold, let us call it the “basic decency level”, you should be denied any kind of reward.

I honestly unsubbed from WoW and switched exclusively to playing this game to get away from that. While the system is acceptable for FATEs, which just award a nominal amount of gil and XP to random individuals and/or groups that spontaneously form and disband, it would turn the duty roulette into a cesspool of elitism and toxicity: "Kick player X because if we wipe one more time, we're missing out on our bonus." If you want to see the system in action, just play a month of WoW.

On the surface the proposal sounds perfectly reasonable, but its foundation is problematic: "If you don’t pass a certain threshold, let us call it the “basic decency level”, you should be denied any kind of reward." How do you judge the "basic decency level?" Some jobs do better on certain fights than others. Some jobs simply suck at certain levels. I was doing on-level FATEs in the Sea of Clouds as a paladin, for instance, and while I was getting full credit on the AoE FATEs, I wasn't getting full credit on the boss ones, despite tanking them exclusively. It wasn't for lack of trying. I just couldn't hold a candle to the DPS of a synced dancer in 270 gear with my 145 item level paladin.

Also, the current system already rewards you for performance in the way of commendations. If you're really killing it and you're still not getting commendations, then I recommend that you stick around at the end of instances until everyone finishes watching their cutscenes so that you can get their vote. The OP is proposing to fix something that's not broken, and his fix goes way beyond alleviating the legitimate issue that you described for a premade group. The situation you described is easily resolved by booting the tank and finding another.