Regardless, I'm not a fan of spellcaster players having access to and being able to pick whatever spells they want.
This is a hill I will gladly die on to disagree.
Back in AD&D I played a Magic User in a six month campaign where we got up to 5th level. During that time, I never found any 2nd level scrolls or spellbooks. DM assured me they were around, we just never made the right choices to come upon them. Upcasting didn't even exist back then, and my character had the power level of a 2nd level character for months and months.
That's completely freaking ridiculous.
This is one of the small but required changes thaty should never be rolled back. Wizards are competent, Intelligent people focused on the Arcane, they can work out two formulas every few months from their readings, experimentations, seeing people cast, and the like so it doesn't even have versimlitude going for it. But from making this a playable game perspective it is absolutely required to be in the base rules because different DMs will have different ideas (just like my DM did back then) and it really makes the game unfun.
If you want to make it a house rule and your players trust you to give them out, that's fine. I'd even play with you. Limit spell selection like the OP suggested but still have free spells per level? Easy. But when working across the gamut of all DMs regardless of experience and of the adventures they are running this absolutely needs to be a rule in the books.
Summation: Because not all DMs run the same, to preserve tthe wizard being viable from a game perspective, Wizards are required to increase spells known outside what they find. All of the other classes can.