Comparing an mmo with a single player game isn't really appropriate, you should compare tank in FFXIV with tanks in FFXI. The whole system is based on the trinity that was established for MMO and has been pretty much standardized since the age of everquest.
If you want the game to be a trinity game it should follow those standards. And in those standards tanks don't do damage. DPS do.
If you want to have a hybrid system, that is fine too. But this isn't a hybrid system, it's supposed to be a trinity one.
As for what makes a tank, is the core tenets of the class: Mitigation and enmity. What a tanks need to do is take the least damage possible and keep the boss/adds on itself. That is the very definition of an mmo tank.
Tanks shouldn't have the same cookie-cutter defensive kit (an 20% cooldown, a 30% cooldown, a short cooldown, an invlu cooldown etc). Tanks defensive kit now is a just a copy/paste with a few changes on top of it. Tanks should also have a slew of abilities concerning gaining and maintaining enmity. There used a be a dps combo and skills which specifically increased enmity, now there are none, it's just a toggle.
Maximizing damage become a thing in HW because the dps checks were too tight. Before that tanks actually melded parry and vit. What happened in HW is that SE made savage raids dps so tight that it made it impossible to clear unless you maximized damage everywhere (including tanks). And you still couldn't clear if you anyone died once. After that it became about "maximizing damage".
During HW it was also the age when tanks had the most distinctive defensive play-style. What killed that defensive play-style was, again, raids. In the raids there was only magical damage, so PLD became the defacto "detrimental" tank since it had neither good damage, nor the defensive kit to deal with magical damage.
All these side effects from raiding trickled down to the general perception resulting in an increased difficulty for the normal playerbase, causing a general perception that tanking was hard and, as a result a decrease in tanks.
It was expected for people to tank in dps stance => people couldn't keep enmity / tanking was more complex/hard => less tanks.
It was expected for people to play DRK or WAR => people had to play the most complex tanks => people found taking difficuly => less tanks.
This resulted in tanks being seen as hard and complicated which caused the subsequent dumbing down in the next two expansions.
SE wants tanks to be more of a traditional mmo tank (rather than a dps), but at the same time they want to keep a sizable population of tanks, which resulted in curbing tank damage throughout the expansions and making tanks simpler.
It's obvious that SE doesn't want tanks to do damage from all the steps they have taken during the years to curb tank damage (like locking accessories, removing main stat materia melds, and the tank stance changes), so asking for more damage isn't really going to wield any results. Even if you find a way to do more damage it will just get nerfed in the next expansion.
The underlying issue with the tanks isn't the damage output. It's the fact that they are too homogenized.