The greater restriction on wizards is prepared spells. You could give them every spell in the game for their spellbook and it will not make them overpowered. They can't prepare enough to make it matter too much.
This. Why do so many DMs think giving more spells to PC wizards will "make them OP"? Complete nonsense. I want to know where this assumption comes from, because I see it all over the place. But consider:
Completely on their own, a level 20 wizard will have 42 non-cantrip spells (anyone else think that number was deliberate? =). Those 42 spells are going to be, in the wizard's calculation, the absolute best ~4 spells for each level. Ipso facto, any spells you give them are going to be,
at best, ~5th from the top of their ranking, if not much much lower.
Furthermore, an optimized level 20 wizard can only prepare 25 of their spells (generally a wizard can prepare about half of what they get automatically, at any level). Unless it's for the fun of variety, a wizard isn't going to skip out on their naturally-acquired top ~4 spells per level when they can only prepare ~2 of them anyway.
Which is really what the perceived power of new spells comes down to. If you as the DM decide to bring enemies with specific vulnerabilities or resistances to spells you know the wizard has,
and you tell the player to be prepared for that well in advance, then yes there will be some edge cases where more spells means more choosing power and better preparation. But a good wizard has already selected a variety of spells to allow them to cover as many scenarios as possible. Wizards don't go on an adventure having only prepared spells that deal one damage type. Many of the best wizard spells don't even deal damage directly anyway.
Your wizard player chose the one class in the game that can pick up more spells than they can use. That's what makes them happy, and it doesn't make them more powerful unless you go out of your way to make it that way... so please, just let your wizards learn more spells.