I can't think of any "modern MMO" game where the tank is the dedicated or designated puller. In modern games, it literally does not matter who pulls. This includes all of WoW for its entire history and all of FF14 for its entire history. It does not and never has mattered who pulls. It may have taken more than literally one input to secure aggro on mobs in earlier versions of these games but it was still not the tank's "job" to pull. It was whoever felt confident doing so. Indeed, at times in WoW, it was sometimes preferable for a ranged class to pull because their attacks or spells had a longer reach than the tank's ranged attacks or pulls. Or because a Hunter could, for example, use Misdirection to apply their aggro to the tank while they pulled the monsters, which ensured they'd go right for the tank. But in no case was there ever a "designated puller" for content.
For older MMOs like FF11, EQ, etc - it was frequently that a tank was *NOT* the "designated puller" for one reason or another. Sometimes they were, sometimes they weren't.
Regardless, YPYT has always been a toxic mentality and behavior that's largely grounded in egotism, not any kind of reality. It happens in FF14 because Square-Enix goes out of their way to try and make each and every tank feel like they're special boys and girls and they're making a big sacrifice to play such a hard and demanding job... but that couldn't be further from the truth. Tank is unquestionably the easiest role in the game to play, especially above dungeons/normals, where DPS tend to just outright die if they get hit by things they ain't supposed to get hit by, and enrage timers mean that suddenly you can't really get away with playing your freestyle SAM anymore.
I wasn't around for it, but I'm lead to believe that tanking was a bit trickier in HW, so maybe that's why they felt the need to add mounts and special ego-stroking achievements? I played WAR during ARR and it wasn't particularly challenging to maintain aggro and stay alive, but I've heard HW made things messy by adding a DPS stance to tanks.
Not really? Aggro is very basic and easy to understand. If the archer hits a monster and pulls a pack, all of those monsters will default to chasing the archer. If the healer does any healing after this point, they will probably swap to the healer. In any case, as long as the archer or healer are in a direct line between those monsters and the tank, they will get scooped up by the tank automatically because the tank is spamming AOEs the whole time.
And even if the tank misses one, monsters in dungeons/normals hit like wet noodles. You can quite literally clear dungeons without a tank, and this is probably by design. I've cleared plenty of dungeons by healer-tanking them after a tank gets disconnected mid-pull and no one felt like waiting for them.