The mismatch between class specific abilities

Barbarian rage: Almost follows the pattern of "number of rage per day = to proficiency bonus". Gets 1 rage per short rest

Bardic inspiration: Number of times per day = charisma modifier. Level 5 = refresh all on short rest

Cleric channel divinity: Starts on level 2. Two uses per day, refreshing one per short rest. 

Druid wild shape: Starts on level 2. Same as cleric

Fighter action surge: Starts on level 2. Once per short rest.

Monk focus: Starts on level 2. Regain all focus points on short rest.

Paladin channel divinity: Starts on level 3. Otherwise same as cleric.

Ranger favored enemy: Follows proficiency bonus exactly, only refreshes on long rest

Rogues: Resource limited actions? No! Unlimited ones thank you very much.

Sorcerer innate sorcery: 2 times a day, refresh on long rest

Warlock pact magic: Almost "number of times per short rest = proficiency bonus -2". 

Wizard arcane recovery: 1 per long rest

I know there are some additional options for some classes at higher levels - but let's ignore that for now and just focus on the early levels for these staple class abilities.Why isn't the same system used for all of them? Do a "number of times per day = proficiency bonus + 1 regained per short rest". 

Is it perhaps because it would make some classes a lot more powerful? Yeah, then the problem is that the ability is too powerful and you have to hack a limit to the ability rather than having a unified mechanic to use that ability. 

So my question would be. Can you see any obvious problems with house ruling all of these class abilities to be = proficiency bonus per day + 1 regained per short rest?

Why isn't the same system used for all of them? Do a "number of times per day = proficiency bonus + 1 regained per short rest". 

Because there's no reason to, especially given each class uses their limited resources in a different manner.

You're falling into the symmetry trap—a very common, almost intuition level belief that symmetric design is somehow better/right.

It's not a hack to dial in the limit of a resource to fit it's power level, it's good design. What isn't good design is coming up with identical resource limits based on the arbitrary reason of "that way they're all the same" and then trying to dial in the power level to fit.

So my question would be. Can you see any obvious problems with house ruling all of these class abilities to be = proficiency bonus per day + 1 regained per short rest?

Yes, these abilities aren't designed to work that way so there's no rational reason to make the change and you'll just risk breaking some of the abilities.

Mentioning that the abilities themselves aren't necessarily of equal merit(as noted above),  although that is somewhat intuitive to some, is still worth stating specifically.

A Barbarians Rage is pretty core to their design, while Bardic Inspiration might be considered this as well BUT, it is a party support utility. Some of these class specific abilities aren't critical core components of the class but, are quality of life or conveniences. Sometimes the benefit is more specific to the character and other times it's a full party buff that is a bigger gain for others or the party at large than it is to the character that invokes the effect.

Remember how everyone hates 4e, partially because of this idea? It tends to make things boring, and that forces all of the abilities to have the exact same power level. I want classes to feel different.

That was actually a criticism of 4e I personally agreed with. I'm going to be simplistic (and a touch hyperbolic) but it'd basically be

"Oh, we're level 4. My wizard gets Forking Blast which lets me deal 2d6 damage to a creature within 10 squares and then 1d6 to a creature within 5 squares of the first. Hey Sarah, what does your Ranger get?"

"She get's a cool Ranger power called Ricochet Arrow. It let's her shoot an enemy within 10 squares to deal 2d6 damage and then bounce the arrow off them to deal 1d6 damage to a second creature within 5 squares..."

"Oh, that's cool, uh Becks, what does your Fighter get?"

"Huh, he gets "Boomerang Throw" which lets him throw a melee weapon to deal 2d6 damage and....yeah...deal another 1d6 to a second creature no more than 5 squares away..."

Now it wasn't literally that bad, but it was pretty darn close. Asymmetry between classes is a strength

Doing something proficiency bonus times per day gets real weird with multiclassing. As PB scales with character level, you have someone with a 1-level dip in a class being able to do the thing as often as someone who is single-classed all the way to 20.

Also, to the OP’s idea that allowing someone to use an ability more often would make the ability more powerful seems like a strange flaw in logic. Letting the character do it more would make it too powerful, so the game restricts how often they can do it. And then you criticize the game for restricting how often they can do it.

It seems like the developers already found the solution to the ability being overpowered, and then you want to remove that solution and blame them for designing the power incorrectly.

If i could wildshape my proficiency bonus number of times, my druid would be such a menace to the table, because wildshape as a feature is so much more varied and potent than say, Bardic inspiration.

The limitations are there to balance out the usefulness of the power. If they were all the same, then classes would lose their flare.  
There was already a rebalancing in 5.5 by shuffling around when Warlock and Cleric get their specialties. They used to get them at level 1, now it is level 3 like everyone else. If they made class progression all the same, it would strip a lot of variety out of the game. 

This honestly makes me think of someone complaining that a recipe calls for a teaspoon of salt, a pinch of chilli powder, and a tablespoon of mixed herbs, with their solution being "Make it a teaspoon of each thing, and if that ruins the recipe, well it's a bad recipe and you should just change the ingredient"

Quote from Maruntoryx >>

Remember how everyone hates 4e, partially because of this idea? It tends to make things boring, and that forces all of the abilities to have the exact same power level. I want classes to feel different.

 So so so much this. I have a huge collection of TSR/WotC books all the way back to 1st edition AD&D, but I took a couple of looks at the 4E PHB and never spent a dime on the entire edition. If DnD goes back to this count me out.

Yes this makes it hard for everything to feel 100% balanced all the time, but this isn't a competitive game, it's all about fun storytelling. "Ooh, we can all do the same thing but it has a different name." - that's 0% interesting. Also not what the OP was suggesting, but bottom line: making classes feel more same-y is definitely not a good thing.

I'd much rather that different characters have different cadences so each one shines at different times and gets their spotlight. Everyone having the spotlight at the same time just means no one gets it.

Now that subclass progression cadence is that same, you need to retain variety in power and frequency of use for abilities. One person gets a powerful attack that they can only do a couple of times, while another gets one that is less powerful but can do half a dozen times. That makes the spotlight shift between them and provides variety.

If anything, I want less uniformity, not more.

I prefer class features to be

(some number based on class level) per day

That way, you dont get a reward for a 1 level dip into a class that uses proficiency bonus to scale that feature.

Sneak Attack is roughly (rogue level/2) d6

If it was (pb)d6, everyone would dip into rogue 1 level to get sneak attack and then use pb to scale the damage.

There's another detail you're missing here. You're cherry-picking abilities to make a point. Other abilities:

Barbarian: Reckless Attack (unlimited uses)
Fighter: Second Wind (start with 2 uses, more over time, recover 1 use on short rest)
Monk: Deflect Attacks (unlimited uses)
Paladin: Lay on Hands (5xPaladinLevel points, use any number you want each time)
Sorcerer: Sorcery Points (SorcererLevel points, used for multiple abilities, Metamagic, and spell slots)

All of this doesn't even touch on subclasses (which is where most Proficiency-based tracking is for abilities), Weapon Mastery as class features enhancing flexibility, or spellcasting as a primary feature of a class. And you're really reaching on Warlock. That tracking only follows for levels 9-10 then 13-20. "Almost" is apparently half, and not the more commonly played half.