You are what you do. In order to become a wizard you more or less need to have devoted yourself to book learning of magic. You can't play a wizard without a spellbook; class has IME the single strongest influence on character because it's what you've decided to do while background is who you were born.
And this is part of the problem with wizard. It's flavour but not terribly interesting flavour.
Hrm?
What are you all doing with your Wizards?
I play since 2018 and so far I have played 4 Wizards. 3 in campaigns and one in a One-shot.
They all played different, felt different and worked different.
My first Wizard was Tas'Ha, a female human evoker, who was an already experienced adventurer, in lost mines of phandelver. She left home, because she didn't want to get married off by her parents and instead wanted a life of adventure. I brought her in to replace my life cleric who was very ineffective (and the healing aspect was moot, because we also had a Bard, a ranger and a Druid...). She was a bad ass adventurer, helping out this group of Inexperienced adventurers, blasting away thanks to the sculpt spell feature.
My next Wizard for a Mini Campaign was Mel'C'ia - she was a 16 year old, living in the slums. She found a spell book on a corpse and started teaching herself wizardry from that (because for that reason I rolled randomly, which spells she had, which lead me to have barley any damaging spells and non of the good damaging spells). Her single mother was trying to keep the family aflot, but her mother got sick and now Mel'C'ia had to keep the family alive and take care of her many siblings. As an illusion Wizards she did street performances with her magic while her brothers and sister pickpocket the distracted viewers.
In the campaign she used her limited randomly selected spell selection (which included the better illusion spells) quit well (thanks to the DM). The campaign/adventure was to find the kidnapped daughter of a high official of the city, which she and the other adventurers where hired to do, which they succeeded in.
During the adventure and at the end, she got her mother healed, an apprenticeship with a real Wizard, a good job for her mother (as the housekeeper of the real wizard) including a good home for her siblings. So far she is one of my most favourite characters I ever played and her ending in that campaign is the most wholesome and fulfilling for any of my characters.
For a OneShot (Lvl.8) I had a Bladesinger 5 / Barbarian 3 Multiclass - she was an experienced adventurerer (like a brawly version of Indian Jones) who spread the word of Lrak Xram, a philosoph who propagates a classless society and revolution against the elites.
I actually just wanted to get high AC with unarmoured defense and the Bladesong, taken together with rage. and of course: Muscle Wizard!
Because of the combination she had some Melee abilities and versatility in dungeon delving. I think with that combo I just reinvented a frankensteined form of a ranger class. She was fun for the OneShot. But I think Level 8 was the sweet spot. Because of the not very synergised classes, on lower levels she would be ineffective and on higher levels fallen behind the power curve.
Right now I'm playing an order of the scribe Wizard. The most typical Wizard. Her Name is Ari, she is a student of Strixhaven (as her background). She actually just wanted to study, be a book worm, do theory crafting and never leave the Univeristy with its beautiful libraries, but Strixhaven University "forces" all of their students to do an internship in the real world.
So she half-heartedly send out some application letter, hoping that nobody would take her. Because she didn't take good care with the applications, she send one of her applications to a Thieves Guild "the Foxes" instead of the the Wizard Albrecht "The Fox" Firethrower. And the Thieves Guild accepted her application (the only ones who did).
She tried to talk to Strixhaven administration to get out of that internship, but her supervisor decided it would do her good to do that internship. She just has to apply her magic while working for them.
That's how the ongoing campaign started for Ari.
So she started her Internship, learned the trades of thieving and thanks to her magic is quite an effective thief, too. It also helps that the gold she makes can help her pay for all the spell scrolls she wants. The DM even allowed me to create my own spell (level 1 spell to create familiars out of paper, 1hp, keep away from water and fire).
That campaign is ongoing and scared poor Ari already a little. Right now she is helping to overthrow the Government of the City State they are in.
Also because it is quite a dense city, she can't really use a lot of fireball, because wooden burnable houses are everywhere. But as an order of scribe she can throw psychballs or iceballs.
For the next levels I will get her the skilled feat so she can get expertise in deception and bump her charisma to 12. Combined with alter self and a cloak of many fashions, she will be quite an effective thief and infiltrator.
So far all my Wizards have played completely differently.
And I promise, I play other classes, too, like a celestial Warlock who pretends to be Bard because she was embarressed that her patron was a unicorn and not a cool demon or devil ...
What I miss for my Wizards so far in the games I played:
Official rules for creating your own spells. The UA was a step in the right direction. But they should be a class feature of the wizard, not a spell.
Official detailed rules for crafting magic items. I know there are barebone ones in Xanathars, but they lack any detail that would make them applicable in the game.
I never felt, that my Wizards were boring, lacked options or were outshone by other casters.
I felt that way with my Life Cleric, because all the other casters in that group had healing, too, and in a fight it doesn't really matter if you get 1 or 6 HP back to get up from unconscious and outside of fights you have hit dice.