What's wrong with bailing out at the first wipe? What makes this different than agreeing to Vote Abandon?
Honestly, leaving the duty and taking the penalty is the best solution in this situation. If someone personally can't deal with a situation, they shouldn't make the other person have to re-queue. Whoever doesn't like what's going on bails the duty.
It's like capitalism. If someone doesn't like the product, they leave it and move on. The party replaces the member with someone who is willing to deal with the problem and keeps going.
The only problem with this is that the duty finder is broken because there's an imbalance of roles.
There's an imbalance of roles because people perceive there to be a marked difference in potential pressure and shame in being a tank, healer, or DPS.
There's a differential in that pressure primarily because of the game's basic structure. For example, it isn't as apparent when DPS is doing a bad job (because performance overall is not visible in-game). Being able to see how good someone's damage is would go a long way in correcting for the DPS roles imbalance.
What's another contributing factor in the difference in pressure placed among the roles? The lack of a culture promoting research and knowledge about your role and duty mechanics. There would be fewer incidents like the one mentioned in OP because people came in to each duty with at least a certain minimum of knowledge and ability, and therefore there would be fewer arguments and less pressure and shame placed on others.
I'm not saying that this is the end-all-be-all solution. I'm suggesting that we can help to partially alleviate the situation.
I'm glad the OP is responding. Not to call you out or anything, since it isn't your fault, but rather the game's fault for not instructing players to do research and learn more before spitting them into Duty Finder and Satasha with hardly an inkling about party play before that. It's also the community's fault for not fostering a more educational culture. Players should know to prepare themselves before they start attempting party duties. It's seriously a sharp curve from the easy-breezy solo play until level 15 and then a full multi-level hour-long dungeon of Satasha and beyond.
Players shouldn't be feeling blindly in the dark at the expense of others. In addition, and importantly, joining the Duty Finder doesn't necessarily mean you're willing to walk someone through their role and the duty if they're clueless. Given the abysmally low populations within some servers (and sometimes within entire Data Centers), Duty Finder is almost a necessity and Party Finder is simply not a very feasible option.
We need to be very honest with ourselves about how much we're asking of experienced players to take someone through their very first dungeon or take someone who has little to no idea what is going on. If this game were truly that simple and mindless, why can't they just figure out for themselves? It's precisely because things are rarely intuitive, difficult to explain, and often "you'll know what I'm talking about when you see it" that makes it necessary to direct players to out-of-game resources. (or, if we really care about immersion, for SE to develop much more comprehensive Active Help information and tutorials for each role). I'm sure other MMO veterans can list off games that do a decent job of taking players step-by-step through their role through quests, as they level, or separate tutorials. I think we can all agree that FFXIV's class and job quests require ZERO knowledge about your role in order to clear, and rarely, if ever, tell you how to act in a party.