OKAY whoops, I looked over the PHB and saw the rates for learning a new skill. 250 days and I GP per day? No way. When you're learning a single new skill and going to class for at least 6 hours a day, it's not going to take more than half a year to learn that skill and it's going to cost WAY more than a couple of magnifying glasses and a spellbook. (Seriously, who decided those prices?)

So, your character wouldn't be going to a typical school or university, they'd be going to a trade school which are very pricey but are also way more instructive. Trade schools don't have to teach a curriculum at a set pace, more gifted students are allowed to move ahead from the rest of the class because the students aren't paying to have their time wasted waiting for slower students to catch up.

In this case, players who want to send their character to school will pay 250 GP a month and each month they're in school they make 4 skill checks with applicable modifiers against a DC 20. Each success deducts 2 from the DC for the next month and reduces the span of time in class by 1 week to a minimum of 1 week intervals, until the player makes all 4 successful skill checks, at which point they gain +1 proficiency and graduate from the course. The higher the proficiency, the more a school will cost as they will be training with exponentially more skilled teachers.
Natural 20's reduce the DC by 4 and count as 2 successes.

So, let's say we have a bard wanting to increase their proficiency in performance to make up for their not INCREDIBLE charisma score, and has STR: 11  DEX:9    CON:11    INT:12    WIS:13    CHA:12 

They could use their superior INT to help them make their monthly skill checks.

Month 1: 15    19     15     15    1 success, DC is now 18.
Month 2: (After 3 weeks): 6     11     14     17    1 success, DC is now 16, next roll is 2 weeks after that.
2 weeks later: 7     7     11     7    bombed out.
2 weeks later (Now 3/4 through month 3): 12     22     16     14    1 success,  DC is now 14.
End of month 3: 12     15     10     9    2 successes, DC is now 10.
Month 3, end of week 1: 19     8     21     22    1 success, DC is now 8.
Month 3, end of week 2: 18     20     4     6    1 success, DC is now 6.
Month 3, end of week 3: 22     14     18     19    Graduated, bard now has reflective shots. Err, an additional +1 proficiency to performance.

Mmmm, that simulation didn't feel great. Let's try this:

Same deal as before, but rather than making 4 rolls a month and staying enrolled until graduated, the course lasts for 4 months. At the end of each month, you roll 5 times. Each success reduces the DC by 2 for the next month. If you succeed 3 times, you graduate. BAM. Done. Money saved. At the end of the 4 months, you get ONE more chance to roll 3 times, and if you can't make your 3 successes, your character has failed the course and will have to try again. The brutal reality of for-profit education.

So,
Month 1: 16     13     8     14     12    1 success, DC is now 18.
Month 2: 13     11     4     13     6    2 successes, DC is now 14.
Month 3: 9     9     12     8     16    graduation! Bard is granted +1 proficiency to the tune of 750 GP, or the cost of a new suit of half plate armor. That seems about right. If the bard was learning say, a brand new language, they would emerge with a full +2 proficiency upon graduation, but improving a learned skill will only award +1 since that's basically paying for a level up.

Also, as a safeguard against abuse, you may not increase your proficiency more than 2 points higher than your current level, so level 1 characters who get a lucky break and have an opportunity to train for a while can only get +3.

I think this should be balanced enough to give players just a little more flexibility with their skills and character building, either improving upon definite weaknesses or amplifying existing strength... or just rounding out a character a little more nicely.

And before anyone says, yeah, that means that you could have a level 10 character with an utterly ABSURD +12 proficiency, but in order to find ANYONE with proficiency 2 points above your own to train, you'd be having to study with gods or ancient spirits (Which, fair play, can happen. It's D&D, after all.) , and personal practice would become practically futile after +5 proficiency. And schools? Teaching legendary heroes? Not likely.

In all, training would probably end up happening very infrequently and by and large won't amount to any more than 1 or 2 extra proficiency points or extra learned skills for the average player, certainly not enough to break the game.

So, INT can be subbed in for the appropriate ability score should their INT be their best attribute, but otherwise, it's not a prerequisite to fast learning. Same with practice.
I was going to mull over group practice? But at that point, that's tutoring unless you have 2 characters both practicing the same skill, which frankly isn't really likely.
I think the school system is sufficiently balanced and simple and clearly the most costly option and definitely aimed for 'inbetween adventure' times.

Thoughts?

I mean- in theory, sure. I think this is much closer to something balanced. At this point in the narrative, a silver piece is a huge gain so it’s hard to wrap my head around 750 gold.  

Don't have a horse in this race, but Xanathar's Guide to Everything has some variant downtime rules for training in a tool or language, which I recall make use of INT to some degree. That may work as a reference.

Those are a little better than the base handbook rules, but for learning brand new skills outside of levelling up, I think the homebrew might just be good enough. Again though, probably will need to use them in a campaign to get some real-world results to see if they need adjustment.

Quote from KinaJardine >>

Those are a little better than the base handbook rules, but for learning brand new skills outside of levelling up, I think the homebrew might just be good enough. Again though, probably will need to use them in a campaign to get some real-world results to see if they need adjustment.

Remember that those rules are for learning languages and tools, not skills.  That difference shows up both in the PHB and XGtE.  I don't think it's a mistake or an oversight.  Learning a new skill like medicine or arcana (given D&D rules and system) is a whole different kettle of fish.

I know, I was more referring to the variant rules stating that training takes 250 days rather than 10 workweeks. A month and a half is way more plausible.

Quote from KinaJardine >>

I know, I was more referring to the variant rules stating that training takes 250 days rather than 10 workweeks. A month and a half is way more plausible.

It depends on what you're trying to learn.  Languages don't really learn well in 10 weeks unless you have some affinity or are completely immersed, for example.

Agreed. Like, I have some thematic solutions to that, but unless a language has been specifically designed to be easy to learn, born from a linguistic system intended to leave as few people as possible illiterate, it realistically should take years to learn a new language and be fluent in it. Even six months would be pretty damn optimistic for being able to converse with locals.

Like, what I would do if a player wants to learn a new language, every reputable teacher would have a supply of an herbal tea that would induce an unnatural state of elevated awareness, allowing them to absorb a foreign vocabulary much faster, so they mostly only have to puzzle out the grammatical structure of the language.

Given how many languages there would be in a fantasy world where some species would be capable of certain sounds that would be utterly impossible for others to make AND would vary from region to region, you'd almost have to rely on some kind of magic to have any hope of communicating with each other. A fantastical world would need fantastical solutions to everyday problems.