A wizard academy… shouldn’t let a wizards be a level 20 adventurer. Nor should an adventurer get all these powers in a month.
I think that levels are Adventuring levels. A wizard at an academy should be able to do ~4th-5th level spells, but not quickly, and maybe not many times per day.
An adventurer is a specific kind of person – a soldier is trained to work in a large group. Even an elite soldier will have a skillset that may not lend itself to adventuring the same way a dedicated adventurer might.
An adventurer is someone who often goes it alone and has to be self reliable. I imagine a wizard can probably exert themselves and cast spells of a higher level, but it takes ages. Or rather, only adventurers have that kind of spell slot stuff.
I think of it like this:
Levels 1-3: Converting your previous experiences into an adventuring-usable ability. For example, the wizard was always focused on abjuration, they just learn how to utilize that in an adventuring environment at 2nd level.
These levels are the conversion levels, where adventuring abilities are largely just learning how to use current skills in this new environment.
Levels 4-8(ish): This is the PCs honing in on the skills they already learned to utilize them as best they can. It is reasonable for a 7th level fighter to not be as skilled with a sword as a number of other masters, or even the city guard’s finest.
These levels are primarily the levels where a PC is actively learning from a master, a manual, or even just watching people. Around level 8 is the limit that conventional knowledge will get you.
Levels 9-14 are mostly building up personal skill. There are few people who can utilize that skill effectively and be able to teach it, so this is largely the PCs learning as they go. This is them combining previous teachings with the knowledge of masters, as well as forging new techniques and skills specifically suited to the PC.
This is about the limit that any kind of (mortal) teaching alone can get you.
Levels 15-20 are largely the PCs forging their own paths, and becoming the all-time masters themselves. The only teaching of these skills might come from forgotten tomes or eldritch beings, but most will forge their own abilities, creating things nobody knows about, combining peak physical power with ability.
At this level, there is almost no one who can train you or make you stronger. Maybe gods or archdevils/fey/etc, but in general, you are blazing your own, personal path.
I also try to implement time skips and longer long rests (gritty realism) to give that time to train and improve. Even if not, the party can use time skips to theorize or train new skills, but only have the ability to reliably use it in an adventure after certain moments (level ups)
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