No, other game simply resort to DPS burns, and when everyone can do damage, then there is no tank or healer, you just rely on consumables. Your wallet size ends up being the only obstacle to overcoming content.
This game has the Tank-Healer-DPS system because the dungeon crawls are designed to not be beaten by consumables, and are thus level synced so that the difficulty is stable throughout the game's life. I will not say the trinity is better because I honestly did prefer Mabinogi's class-less system before they went and ruined it with classes.
A game has to be designed from the beginning to have, or not have these things. FFXIV originally did NOT have these. Read the original skill lists for for 1.00.
The problem is not that the roles exist, the problem is that people come to the game and have an entitled world view that they should be able to play the content without any teamwork. Hence you have annoying stupid things like green dps and ice mages because players want to play it in a way that it was not supposed to be played.
If the trinity was actually working the way it was intended, tanks would have no reason to turn tank stance off, the second they did, the healer would rip all the enmity from them. If healers were playing it the way it was intended, they would be non-stop casting healing, and the second they stopped, the party dies. Neither of these happen because that would make the game too punishing to the "I just want to be a blue/green dps" people. Like relative to the beginning of 2.0 both healers and tanks have been nerfed to basically being unnecessary for all but the highest tier content. PotD basically made it so that nobody even needs to learn to tank or heal, here you go, go be a green dps/blue dps/ice mage, your skill level is meaningless. There is no skill involved in zerging content. Just keep mashing the dps button.
The DPS queues remain high because players don't want to do any teamwork that doesn't favor themselves. People who play tanks and healers regularly know how to play those roles on most content. Players that only ever DPS aren't willing to try these roles because they don't do big numbers when they think in terms of DPS. There are also other outside forces at work that ensure that players don't switch to tanks or healers.