I can certainly understand the sentiment of wanting the "customizeable" part of feats available for every character (as an option for those who actually use feats)-- basically a lot of the General category of feats fall into that bucket. But I believed and said many times during 3E *and* 4E that they REALLY needed to split those off from the combat related ones. "Feats and Traits" were the buzzwords of the day. And I STILL believe that should be the case.
Given my druthers... here is what I'd prefer to see for character generation:
1) Choose a Race. For your 'Basic' game, you have your Core Four (human, dwarf, elf, halfling). For Standard games, throw in the halfelf, halfork, gnome, and dragonborn.
2) Choose a Class. For your 'Basic' game, you have your Core Four (Fighter, Rogue, Cleric, Wizard), all of which have a single pre-decided build. For Standard games, throw in the other classes (Druid, Ranger, Paladin, Monk, Barbarian, Bard et. al.), and open up the various builds (Styles, Schemes, Deities, Traditions, Favored Enemies, Oaths etc. etc.)
3) Choose a Background. For your 'Basic' game, you skip this step and instead the Core Four classes each get a bonus to a certain ability check (Fighter/STR, Rogue/DEX, Cleric/CHA, Wizard/INT). For Standard games, you have the long list to choose from each of which gives 4 skills and a special bonus feature.
4) Choose a Trait. This would be the replacement for the General category of Feats. At chargen for Standard games, you choose one (or two) Traits that are created precisely to be useful to almost ALL the classes in some form or fashion. You don't get more over time, instead they just give you a hint of what kind of person you are to begin with. Some Traits might include:
Alert - you cannot be Surprised.
Durable - when you roll Hit Dice to regain HP, roll two and take the higher
Fit - you gain extra HP equal to your level
Fleet - you gain a +4 bonus to Initiative checks
Magical - you gain two cantrips from either Cleric, Wizard, or Druid lists
Reflexive - on your first turn of combat you have Advantage against any enemies not yet acted
Thick-Skinned - you gain a +1 bonus to AC
And you can come up with a bunch others. Basically... Traits should be those abilities that are innate (and thus unlearned) to a person, affecting those parts of the character that do not fall under the purview of the skill system. The same sort of abilities we see assigned as Racial features. But because Traits are an add-on to give an extra bennies (or two) to characters, they do not have to be used in the Basic game.
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And now that you have these four pillars of chargen taken care of... you can move all the "Expert Feats" into a silo for the Rogue as his Class Features (lets call them Knacks the hell of it). All of these Knacks make use of the Skill die, exchanging it for other skill-based abilities. Just like Spells, Knacks are assigned a level of power... and just like spellcasters, Rogues receive 1 or more 1st level Knacks at 1st Level, 2nd level Knacks at 3rd Level, 3rd level Knacks at 5th Level and so on. And for ease-of-use... the game might bundle a set progression of Knacks together into a Rogue Scheme. This Scheme would grant your Knack bundle, plus give a couple extra features-- the same way selecting Deities and Traditions does for the Cleric and Wizard. Your Acrobat, your Assassin, your Thug, your Charlatan etc.
For the Fighters? All the Martial Feats are also silo'd off to become its own group of class features which we call Maneuvers. Like the other Core Four classes... Maneuvers are divided into levels of power, and the Fighter acquires them using the same table of every-other-level. If we keep the idea of Expertise dice, the Fighter spends them to trigger his Maneuvers. And just like the others... the Fighter has several different Styles to select from that bundle a set progression of maneuvers together, plus grants a few additional features for that Style. Your Reaper, Defender, Skirmisher, Sharpshooter etc.
In terms of the Basic game? The Basic Fighter receives a Class Feature every-other-level which is one of the Maneuvers pre-selected (usually the easiest-to-grasp and easiest-to-adjudicate ones). Same thing for the Basic Rogue and pre-selected Knacks.
And finally as I mentioned before... because Fighters, Rogue, Clerics, and Wizards all use the same table for when they acquire a new level of Maneuver, Knack or Spell (and truth be told, the Druid would also fall in here as a full caster as well)... you can mix and match for multi-classing much easier. Plus... you now also have these three silos of Maneuvers, Knacks, and Spells that the other classes can possibly gain access into at a reduced rate of acquisition. So the Monk might be able to gain Maneuvers at 2nd, 6th, 10th, 14th, and 18th, rather than at the Fighter's 1st, 3rd, 5th, 7th, 9th, 11th, 13th, 15th, and 17th. The Monk won't gain full access to ALL of the Maneuvers list obviously (and probably might only have a small select few to choose from if they're not outright pre-selected for him)... but at least the Monk can get some additional martial abilities from an individual combat-focused silo. And the same would hold true for the Barbarian, Paladin, and Ranger.