You have some interesting points, some I agree with and that aren't to my personal taste. Here's some feedback that even where I don't agree may be useful for refining the ideas.
The skill system - I like rule simplicity, but a simple binary system is just not good enough. Proficiency should be tiered. I'd have 4 proficiency levels - non-proficient (as raw), amateur (half prof. bonus), proficient (as raw), and expert (proficiency bonus + 3).
Every campaign I've been in has eneded int he 8-11th range. Say 90% of my total play time is 8th an lower. At hose levels, the extra proficiency bonuses you put in besides expert are using just a +1 because of how low proficiency is. If you are going to add complexity, don't make it for a trivially overlooked adjustment.
I excluded Expert. Comparing it to the current Expertise which double proficiency, this is also in the +/-1 of the current system, except now all the way up to 13th so 100% of mty play experience.
In other words, this doesn't have enough affect on play to justify adding complexity. If you want to add additional tiers, make them meaningful. If they aren't meaningful, you're just complicating the system for little reason.
I also think they really bungled the skills tied to Int and Wis. I'd revise those skills in particular.
These are pretty interesting, and your logic seems sound to me.
Too dump stat oriented - the attributes (other than perhaps Dex) do not do enough outside of class abilities. Some people like this; I don't - I think it makes attributes dull and pointless, and it leads to lack of character diversity within a given class. I'd tie more mechanics to each in an attempt to make all attributes at least somewhat attractive for all classes. Int would grant bonus proficiencies along with Tactics as a useful combat-oriented skill. Charisma would be baked into the Inspiration mechanic as well as Reknown. Encumbrance is too fiddly, so I'd give all weapons and armor a minimum strength requirement for proficiency (this also helps circumvent the paradoxical and hyper-unrealistic "8 strength Longbow master" problem we see in the core rules)
I think you could do a good job in an RPG with these ideas. I don't know if they bolt onto 5e particularly well unless you also change a number of the surrounding affects of those design choices.
Either they all are needed, or you can pick any few. If you can do a high-WIS-only fighter, you've strayed fairly far from D&D sacred cows already, so I'm going to instead assume that you mean all are valuable to characters.
5e doesn't give the ability to advance lots of stats if they all are needed. As a matter of fact,t here's a real faustian choice in 5e about ASI (vs. half-feat) vs. feat, and feats are what give characters choices and make them more interesting. Requiring spending ASI/feat selections more on "math bonuses" to stay relevant will ultimately make characters less interesting and less varied from each other.
If you change it, say giving ASIs that can give +1 to three stats, you have ability score inflation. That doesn't play well with bounded accuracy, it negates the assumptions that everyone will have some Achilles Heels in terms of poor saves, and frankly some classes can really take better advantage of several good ability scores than others, so there would also need to be class rebalancing.
In other words, it doesn't work with the current 5e ability score system, and would take a lot of work on the surroundings to include a more robust ability score increase system.
Rules for small races - Small race rules should be relative, not absolute. A small race in an environment with nothing but small races should not have special rules applied to them. So I would not have rules like "can't use two handed weapons". I would instead have written more comprehensive size rules that are only applied when interacting with targets of different size categories. For example, having Str based attacks do half damage against medium or larger targets, or granting AC bonus against larger targets.
Having a large changes (like half damage) invalidate a bunch of build concepts. And frankly, I'd rather a game allow me my fantasy archetypes instead of penalizing me with too much realism in my escapist fantasy game.
Workable changes make sense, though again make sure they are meaningful changes. Also make sure that they never need to be recalculated in combat regardless if you are fighting a pixie or t-rex - combat is the longest single mechanical activity by the wall clock and introducing additional math comes only with overwhelming new advantages.
Scaling on basic items - Items like Basic Poison and Healing potions should possess a modicum of scaling with level. Not a lot, mind you, but some. A 10th level character using a basic healing potion should healed for moderately more HP than a 1st level character using a basic healing potion. I'd revise the formulas for all these items. Poison would probably damage based on target level/CR, scaling from 1d4 up to 2d6 or something.
Why? Rather, if it scales then everything about it should scale,including cost as a percentage of wealth. If a healing potion is a considerable investment for a 1st level character, a potions that heals the same percentage of HPs for a 5th level character should still be a considerable investment. Now, it may not scale that much, but then the price doesn't scale that much.
Which brings us back to - if the price differs, do we want that there is mysterious inflation because you are higher level, or just more effective (and costly) potions which keep verisimilitude and make narrative sense?
No official scholarly cleric variant - I've always found the lack of a prominent wizardly cleric in D&D weird and offputting, IMO, there needs to be an official variant that turns clerics into d6 HD robe wearers with a lot of scholarly knowledge and a spell list as good as the wizard's.
I'm absolutely fine for this. The 5e design space is easily set up to allow new classes and a non-warrior cleric makes plenty of sense.
No rules for "multiclassing" subclasses - This is a pretty obvious thing that really shouldn't have been overlooked. Subclasses were a great concept, but they should have been designed in a way that allowed for branching and going back within a class to built more complex characters.
I have both thematic and mechanical issues with this.
Thematically, for most classes (*cough*dang-fighter*cough*), the subclasses are specific themes. Swapping back and forth undercuts that.
Mechanically, some features are better than others though it usually averages out in the end. This means that picking and chosing could end up effectively reducing choice as the "best" options per level get picked and characters end up getting more and more alike.
Mechanically it also has the issue that you need mroe playtesting and balancing because you can create more combinations. Oftne this means either some combonations get much more powerful (*cough*3.5*cough*) or everythign gets watered down so the choices can't end up as force multipliers. Neither is a direction I like.
Think about exactly what the design space of subclasses is (compared to classes), and that each provides a distinct flavor and set of mechanics is a real part of that.
4 Attacks for fighters - it feels ridiculous with Action surge. I'd replace fighter's level 20 ability with something equally potent but less conceptually absurd, and ideally something that has a more versatile role, such as a boon for tank fighters.
20th level is a myth for 99% of gamers with the possible exception of a one shot adventure to give it a try. And for the 1% it's not a myth for, I doubt it takes up more than 5% of their total time playing. Do what you want.
As I mentioned originally, some of these I like and some I think need some additional consideration to make them and their repercussions fit into the existing 5e design. Some of those are still ideas I'd like to see, but in an RPG that incorporates them from the ground up.