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