My intent here is to create a wizard healer school that is viable and has a flavor that works. I know there are some who will object to the whole idea. I can’t really convince you if you are strictly against the idea altogether. I can only say that this proscription against wizards who heal is strictly a D&D thing. Plenty of games, and folklore traditions, show less of a separation on this point.
I did try to make the thematics of this work for me. The Vitomancer is a close cousin of the Necromancer but inverts that connection to give us the healing wizard.
School of Vitomancy
Life and death are harder to define than many people think. It is more of a spectrum of phenomenon than many people realize. The Vitomancer recognizes that spectrum and manipulates the energies of life and death together.
They are often misunderstood, painted with the same negative image many people have for Necromancers. The Vitomancer can cause harm but can also heal and it is their choices that prove them good or bad.
Beginning when you select this school at 2nd level, the gold and time you must spend to copy a necromancy spell into your spellbook is halved.
Starting at 2nd level, you begin to pool of life energy pulled from necromantic sources. When you cast a Necromancy spell of 1st level or higher, you can simultaneously use the extracted energy into a healing pool. This healing is equal to your wizard level + your intelligence modifier and can be applied to any target you can touch as an action. Once the pool is expended, it can be restored once more after a short rest.
Additionally, Cure Wounds is added to your spellbook and is considered Necromancy for you.
The Balance Restored
Starting at 6th level, your mastery of life energies has improved further. You can do more than just heal. You can expend 5 hit points from your pool of healing to cure the target of one disease or neutralize one poison affecting it. This can be a single action to remove multiple effects so long as all healing pool expenditures are done at that time.
Additionally, you are immune to diseases.
Drawing Upon Life
At 10th level, you may lower the maximum of your own hit point maximum by 10 to do any of the following effects: (edited based on feedback, and wanting to make sure it is not too unbalanced)
As a reaction, you can redirect your energies into your own constitution allowing you to have advantage on all constitution checks for 1 minute.
As an action, remove a level of exhaustion on yourself or any target you touch.(edited after consultation with B. Stoddard to better reflect the intent)
As a bonus action, regain a spell slot. The spell slot of level that is equal to or less than half your wizard level (rounded up), and not 6th level or higher.
As a reaction, you can prevent a single target within line of sight from regaining hit points in any way till the end of your next turn.
The hit point maximum is restored after a short rest.
Master Life and Death
Starting at 14th level, you have seen the connection between life and death. Any necrotic damage you do is added to your healing pool.
In addition, Raise Dead gets added to your spellbook.
Designer Notes: My intention here is to make this a viable healer but maybe not as good a healer as another class more traditionally attached to that function. A life cleric is just a better healer, by design. I think it provides interesting choices, especially at 10th level when you can start sacrificing health for effects. Given that you can turn around and heal yourself, that may seem less like a sacrifice, but healing takes an action and gaining multiple actions in a turn is hard for wizards.