How would you tackle job identity?

[Shurrikhan]

In wow dungeons certain classes are significantly better than the alternatives and creates scenarios where people just do not accept certain healers, or certain tanks or certain dps.

This no more less true in WoW's Mythic+ even past reward caps than it is in XIV's Savage raids even now, despite the greater homogeneity in the latter.

Yes, if something is flagrantly undertuned due to a terrible tier set or a new bug, people in the know will want more ilvl and/or achievement score from that spec than most (as they may from specs that are rarely well understood except until very high level content, specs that tend to attract "auto-pilot" players, etc.), but until you're deep into "purely for the ego points" territory, everything gets picked so long as the overall comp still covers the basics (a battle res, sufficient displacement for the particular dungeon, a spare dispel for Afflicted weeks, enough CC for incorp weeks, etc.), which is not particularly limiting at all.

Though there are some FotMs in most patches (Vengeance Demon Hunter being a longstanding one this expansion), the reason a given player is declined short of a particular fight of Mythic raiding or the highest tiers of Mythic+ dungeoning... is more often equivalent to "We already have enough of A, B, and C capacities you bring, not enough of E and F, and already have a Physical Ranged for your particular raidbuff" rather than "We don't take your kind."