There is still a world of difference between Hercules and Anime Warrior.
Really? Because Hercules is a demigod that journeyed to the Underworld thrice (returning alive every time, taking Hades's dog Spot with him one time), wrestled Thanatos into submission, was able to hold the entire weight of the sky on his shoulders, killed and
skinned an invulnerable lion, and became a god after he died.
. . . What kind of anime have you seen? Because Hercules is pretty comparable to the warriors from the few anime I've seen.
One of the problems (IMO) of 5th Ed is too much magic. Every class or race has it in some form.
Which IMO takes away from those classes that are supposed to be the magic uses.
Same issue I had with 4th Ed. Everyone could heal. Every class had one or more attacks that was “Deal X damage to target and heal X damage to yourself.”
1) There are magical options for every class (and, consequentially race, because any class can have magic). Doesn't mean that you have to pick them or that spellcasters are "watered down" because you can cast a couple cantrips and maybe the equivalent of a 2nd-4th level spell.
2) Not all healing (even "spontaneous healing") has to be "magical". Hit points are an abstraction of physical strength and endurance, will to live, agility, and luck. So, therefore, a nonmagical "inspiring" ability that increases your "will to live" or something like that could be healing. I don't have a problem with every class having some healing ability because hit points aren't just meat points.
As you said though modern pop culture has to bleed in. And them youngsters today love their Dragonballs and Pokemons and One Pieces.

. Though I see it more as “Superhero” then “Anime” but that’s just my grumpy old man POV.
So to Be modern and appeal to the new crowd fighters and rogues need to be able to teleport or turn invisible or magically produce weapons or whatever.
Meanwhile I’ll go back to an older ed for a bit until I’ve had enough “normal” then maybe try D&Ds take on the Superhero/Anime Fantasy Genre.
Young people (in general) like playing powerful heroes and don't care much for the pesky bookkeeping that comes along with keeping track of equipment and that kind of stuff. So, games sometimes cater to that preference by making more abstract versions of older game mechanics, where you can magically summon throwing daggers that disappear when they hit, or have you turn invisible as an abstract stealth mechanic, or have you teleport to emulate the "you're moving really fast" mechanics.