Levitate should not be an Ability:
It is that simple, if a Pokémon is above ground or not, should be something completely seperate from the Ability System.

Give Sound Moves their own Type:
Having most of those as Normal Type doesn’t make much sense, and there is just so many Pokémon and Abilities that are geared towards Sound.
Sound could be super Effective on Psychic (Loud Noise hurts Brain), Ground (Vibrations, Sonar), Fire (Sound can put out Candles) and Water (Sound travels further Underwater).
It could be not very Effective on Flying (Birds constantly have Wind in their Earholes) and Ghost (Maybe even 0 due to originally being Normal Type) Types.
Apart from the obvious ones like Jigglypuff and Voltorb, this could also be a typing for the Whales, Dolphins, Bats and such.

Strength should be a Rock Type Move:
Why? Because you hurl a Boulder at the Enemy, that’s why!

HMs should just be all Field Moves:
First of all, the whole "you cant override an HM Move" shit only exists because you were able to store away HMs in the first few Games, so lets remove that.
Secondly, since the newest Games all replaced HMs with a Phone Service you can call a Pokémon Taxi of sorts with, why not have a Service that rescues you in case of a Softlock, and brings you back to whichever spot the very first Field Move was required to progress (so Teleport/Dig stays useful).
Thirdly now that all obstacles are out of the way, have the term HM just mean "Move you can use outside of Battle", making the term HM just a useful intuitive marking of a Move such as Dig, Teleport, Headbutt and others.
Instead of HM Licenses through Gym Leaders, you just get/find the HMs whenever you require them for the linear Story to progress. Or the Gym Leader gives you an HM and a TM as reward. This should remove the confusion of why you cannot use a Field Move.

Four Moves are not enough, but more Type Coverage is bad too:
Many people including myself hate HMs for wasting valuable Move Slots. So why did I suggest the reintroduction of HMs as Field Moves? Because of this Idea!
Have 6 Move Slots instead of 4, but to prevent increasing Type Coverage of Moves, have Move Slots limited depending on the (Dual-)Type of the Pokemon using them.
First for Mono-Type Pokémon, 3 of your 6 Move Slots have to be the same Type that the Pokémon is. If it is a Pikachu, then there is only 3 Slots usable for non-Electric Moves (Iron Tail, Surf and Quick Attack for example).
This forces you to have 3 Electric Moves, and also causes Move Types for Non-Damaging Utility Moves like Thunderwave to actually make Game Mechanical sense.
And for the Dual-Types, 4 of your 6 Move Slots have to be of one of the Types the Pokémon is. Yes, only 2 Wildcards for this one, to make sure only up to 4 different Move Types can be used.
So if you have a Bibarel, the perfect HM Slave, you get to teach it two more Water/Normal HMs than was possible before.
Or if you don’t want an HM Slave, you will less likely be wasting a good Move Slot on one of your other Pokémon that way, since you are forced to have at least 4 STAB Moves anyways (you can have 6 if you really want to, but that wouldn’t be as useful, would it?).
All of this will make Utility Moves much more useful and less of a waste as well, since you have more Move Slots, but putting more than one Damaging Move per Type does not make much sense, unless you can hit more than one Target with one, while the other is higher Damage, and yet another has a chance to paralyze or so.