Something I noticed recently is that all the spellcasting classes, other than the Wizard, have access to healing spells and/or healing abilities outside of skills. Clerics and Paladins have access to Healing as a default with their spells and abilities (especially with some of their subclasses), Rangers and Druids have access to Healing spells as their spell lists, Warlocks get a Healing ability and access to the Cleric spell list with the Celestial pact subclass, the Sorcerer has the Divine Soul subclass (which gives them access to Cleric spells), and the Artificer even gets access to a few Healing spells. The Wizard is the only casting class that doesn't gain access to Healing spells without either multi-classing and/or taking a feat that gives them access to those sorts of spells.

Some people might say it's part of the design philosophy for class, giving them access to Healing spells could make them potentially too dangerous/powerful. And I can understand that, but Warlock and Sorcerer had similar design philosophies until their subclasses that gave them access to Healing popped up. So, would it really make any difference for a Wizard subclass to have access to Healing at this point? They would still have the limited number of spells they can cast and low Hit Points that they always do, and having a Wizard that doubles as an emergency healer sounds pretty cool to me. And honestly, a subclass like that could just add to the Wizard's versatility, and having a subclass like that would just add in potential spells to agonize over getting for the Wizard. So, everything pretty much stays the same in that regard.

There is an old UA subclass for wizard that never made it to official publication called Theurgy.  You chose a domain and gained domain spells and abilities.....once you learned the domain spells you could learn other cleric spells as wizard spells.

Cool, I must have missed that UA

I didn't notice that, but that's still just a single spell

Although Wizards don’t get the classic direct healing spells, you have some spells to boost that. False Life, Vampiric Touch and Life Transference come to mind.

However, one thing that Wizards have and other classes not that much, is easy access to abuse “summon” action economy in order to consume potions.

Familiars, Unseen Servants, Tiny Servants. Of course you need to buy potions, but you could heal up to three times in a single turn using your own action to get a potion yourself, then using your bonus action to command your Unseen Servant to deliver a potion to you or someone else, and ultimately drive your Familiar to do the same a third time. At level 1… who needs Mass Healing Word, right?

You can also be a Mark of Healing Halfling and get some healing spells as wizard spells that way.

And there are a few feats that give you healing. I personally think those would go better with "emergency healer" than a healing-focused subclass.

And wizards already get some of the best spells in the game, and lots of them, so it would create an unbalance, I think. Plus, if you could heal, you could heal yourself. And that kinda takes away from their squishiness.

I think a major reason, if not the main reason, Wizards don’t get healing spells (unless you include Life Transference) by default is their Level 18 class feature, Signature Spells. An at-willl Cure Wounds or Healing Word effectively creates unlimited healing, which breaks the game even by Tier 4 standards.

I like that book magic doesn't have the potentially wholesome or fulfilling effects of channelled magic.