This is, imho, the most valid argument against parsers. It also ties into the everything becoming a dps race point I made a few pages back.
A counter to this, is to look at the games current meta, everything is dps, tanks, healers, everything is dps centric. Not exactly on you point I agree. Part of the fault in this, and your feeling, lies with the developers I think. Developers tend to consistently fall back on dps, and dps checks as the main core mechanic in almost every encounter you come across. If encounters were designed to be less reliant on dps, then I think it would alleviate this problem. However, this is a problem with MMO's as a whole, not just FFXIV. And meant for another thread I would say.
But, yes, I totally agree with they way developers set up the game, that it can break immersion.
However, I also have only found myself in the same situation after the content is on farm mode. In which case, what you suggested is already happening for other reasons.
On top of this, I feel that as long as the developers continue to add content with tight dps checks, alleviating the problems that come with the constant failures of these instances is of greater importance than the effect of a parser on immersion after content is on farm. Also, once again on the developers, this is currently a hot topic as players can get as far as level 60 content and still be doing little more damage than auto attacking, as commonly linked parsers have shown. You have people doing dps in the 200s, where close to 1000 is very easy to accomplish, and only say 600 is needed to complete almost anything. That is a pretty big discrepancy.
I still contend that the majority of low performing players have no idea that they are performing poorly, and think they are doing enough or more than enough. Therefore I think if they were presented with the actual, they would take steps, however small to improve.