Wizards work fine whether they have the minimum spells in their spellbooks or have every wizard spell available in the game. It really isn't a balance concern, so you should feel free to do whatever will make for the best story.
In a situation like this, I like to look to the real world for inspiration. The closest analogy to spells is computer code. In the real world, you see some people that are willing to share code they write, while others don't want to share their code unless they are compensated - although once someone else gets their hands on your code, it is hard to keep them from spreading it around.
To translate that to the real world, I generally say that anything in the PHB can be found pretty easily as they are common spells that have 'entered the public domain'. I allow wizards to select any PHB spell as a spell they can gain when advancing a level - but in order to select a spell from another source, they have to explain to me where they gained access to the spell. Often this is something we roleplay out. I don't prevent them from selecting the spells, generally, but it sometimes gives us a story hook.
I also add a lot of spells to the game. I've had a book of 'personal' spells that I began in the 80s. I have converted them from AD&D to 2E to 3E to 5E, constantly adding more spells that I create or borrow from 3rd party sources. They're spells that my wizards created, spells that NPC wizards used, and spells that I put on scrolls. I find that having unique spells not found in the core books makes the game feel more immersive. Adding spells with a bit of mystery and uniqueness to them adds a lot to the game and makes it less of a 'cookie cutter' feel and more of a story.
