No more than 4-5 spells, really. Not even "per spell level", but for their entire career.
For a wizard to be effective in D&D, I say 4:
1. Offensive, pure damage dealing, to subdue or kill (possibly secondary effects, like setting things on fire or freezing things). Burning Hands, [the classic] Magic Missile, Fireball, Lightning Bolt, Ice Storm...any one will do.
2. Offensive, to incapacitate. Maybe with minor damage dealing, but essentially to "capture" not kill. Sleep, Charm Person, Web, Hold Person all come to mind.
3. Defensive. I personally prefer your comic book-esque "energy shields/force fields" kind of stuff, your Shield, your Resilient Sphere, etc... But in D&D this could also be any number of illusions: Mirror Image, Invisibility; ceremonial type magic: Protection from Evil, Magic Circle, Banishment; the "Wall" spells; etc...
4. The 4th slot is the one that fluctuates. In 5e, this would be changing from day to day, or used as rituals when the need arises. Utility and divinatory stuff. Your Unseen Servant, Floating Disc, Comprehending Languages, Detecting Magic, Knock/Wizard Lock, Water Breathing...whatever comes up. But, generally speaking, you shouldn't really need to delve into your spellbook/"known list" more than once or twice per day.
Optionally, there are 5.
5. Movement related. Pull Levitate/Fly out of the "utility" spell slot. But, in D&D 5e, you also have your Jump and Longstrider, Misty Step, and, naturally, Dimension Door and Teleport at higher levels. But Levitate/Flight is just vastly utilitarian in any number of exploration and combat situations. Enough so, I think, it could be it's own 5th spell per day.
So, yeah. 1. Offensive. 2. Non-lethal Offense. 3. Defensive. 4. Movement, and 5. Utility "open" slot that can change depending on the situation/day.
If magic is supposed to be more rare and difficult to come by, how on earth would you justify getting/giving 10-12 spells per spell level?! That just sounds like greedy entitled nonsense on the player's part. Even your original number was overly generous, to my mind.