Following up on this idea, I haven’t really played any MOBAs myself, but I did get into Overwatch for a bit, which combines MOBA elements with FPS gameplay, and one thing I noticed is that tanks there do manage aggro, in a sense. It’s just that instead of “agro” being a numerical value that you increase or decrease with your abilities and the computer uses to determine NPC behavior, you manage the other players’ actual attention by selectively applying pressure. The enemy team knows it’s not in their best interest to shoot at you (assuming you’re the tank) because you’re the hardest character on your team to kill. Your job is to pressure them, to make yourself enough of a problem that they can’t afford to ignore you, even though they know that shooting at you is literally less valuable than shooting at your team’s healers or DPS characters. Accordingly, a lot of tank characters are just as offensively powerful as, or even more offensively powerful than, the DPS characters, with DPS characters’ main advantage over tanks actually being their mobility.
4e’s marking mechanics and various Defender powers imitated aggro-as-numerical-value from MMOs. But since that proved unpopular, maybe 5e would do better to embrace the aggro-as-pressure model from MOBAs.