I honestly think they need to bring back all the permanent stuff. Permanent aging from Haste and Wish, permanent changes to ability scores from Books and various other sources, magic locations or complex Elixirs like in the old Volo's Guide to All Things Magical that permanently grant both active and passive abilities. Stuff that makes a given character unique compared to other characters of the same class, level, and starting ability scores.
The tightly limited game engines of 4/5e are not friendly to giving people free templates (what you are talking about is essentially a LA+0 template in 3.5 terms) - It blows up the math if they add numbers in any way.
I don't know what your problem is, honestly. If you want everyone in 5e to have unique powers, have everyone play the casting archetype of their class and
pick different spells
wow, everyone is different now. Amazing!
The best approach if you want everyone to have unique powers is to just do what 4e did and get rid of the caster/martial divide and the frankly shitty legacy code of "spells". - Everyone has powers, with differing mechanics but roughly equal power and breadth - flavor them however you like
Boom, done. It's more balanced
and more "unique"
Arguably the imbalance of 3.5 has it's own charm, e.g. a beginner needs to understand/apply a lot less "lines of rules" to play a fighter compared to any other class.
I may even go so far as to say that D&D is so popular because it's imbalanced, the realizations as you learn the system and go "ohoho so that's possible" are pretty addicting.
we usually use terms like "optimization" or "dollcrafting" to reflect that the difference between a character crafted with High System Mastery is much more potent that one crafted with low understanding of mechanics.
this adds a "skill cap" to the game, and as you say, it can be fun, but it's kinda a headache for the DM, since it makes it much harder to gauge difficulty and provided an appropriate challenge that is neither cakewalk nor TPK.
(also you're wrong in your example - the 3e fighter is extremely frustrating and unrewarding to play as a newbie who doesn't know what feats to pick. The barbarian is a good newbie class, ditto a "blaster caster" - fighter is actually hard to use well.)
it takes a special kind of spreadsheet nerd to like designing
battlemechs character builds, but the simpler game engines of 4/5e are probably going to be easier and more appealing for many.