There are three ways though: ASI to any ability score, feat, and multiclassing. It’s late, I may be forgetting others.
And while yes, most people use an ASI to their primary stat, that’s their choice to do so. The game gives you an option of being better at a pool of skills by applying ASI to any stat. If you dont apply it that way, well, that’s your choice then, but Bob the fighter CAN get better at lore or perception by putting his ASI into the appropriate stat.
ofc, it is their choice...but, if +2 to primary ability is better in 9/10 situations, then there is no reason to have other options.
That is why I am for separate resource pool for feats, skills, tools/languages, ASIs and general pool for anything.
It is always the player's choice. You can max out your strengths or shore up your weaknesses. As a player, I am always happy to shore up any weaknesses my characters have. In 5E I am very happy with my primary stats at 16. It isn't max, obviously, but it is "good enough" that my character can overcome the challenges she faces.
Does she have proficiency in deception? intimidation? Nope. Nope. What about Charisma saves? No, again. So if I find applying my ASI to Charisma (if it is even, like a 10) is a good thing to gain a +1 to saves and skills she wants to be better at. Or if it is odd, I take Resilient and apply it to Charisma, not only gaining a +1 to those skills she is not proficient in but also she gains proficiency in that save, which is great.
Would boosting her Intelligence to 18 be better? Probably since she is a Wizard primarily, but then she never improves in things were she is vulnerable. It all boils down to your philosophy.
Finally, as to the OP, let's take something more real to life for me, personally, and that's Athletics. When I was younger, I did a lot of sports and would probably have had a STR of 12 or 14 for a +1 or +2 mod. Despite being active in sports, would I have had proficiency in it in general? Maybe, but there were a lot of others who were MUCH better than I was so I would be inclined to think I didn't have proficiency if they did, you know?
Well, now I am older and I've learned some new sports over the years, sure, but am I any better at the stuff I used to do? Nope. Why? Because I haven't applied myself to it and practiced over the last 20-25 years. If anything, I am sure my Strength HAS gone down and now would like be only a 10 or maybe a 12, and since I haven't kept up with those sports, I am not even as good as I once was.
If you don't have a system for skills degrading, I don't think it makes sense to have a general system to improve them "overall". At best, I think a skill swapping system makes more sense. Perhaps with each ASI you can swap proficiency from one skill you do know to one skill you don't know. Maybe after a few levels you realize proficiency in Animal Handling hasn't been useful for your Fighter, and wish you had perception? Well, swap them out. This way, you aren't spending resources to learn a new skill or waste an ASI, etc. but it allows you to focus the resources you already have into where you want them.
It would definitely help IMO because I've seen numerous times where players regret skill choices and wish they had others as their characters grow and change.