Wizards not being able to heal in D&D, is the stupidest thing about D&D. Would be awesome if WOC would finally give Wizards ability to heal, since there is NO logical reason for them to not being able to heal, rather than to nerf and "balance" the class. Divine Soul Sorcerers can, Celestial Warlocks can, Bards, Druids, Clerics, dude, what the hell.
No. Wizard's should not heal as base because that means their only weakness as a class would be covered. They are already one of the strongest classes in base game DnD 5e and balence wise it makes no sense. On top of this it has been established in the lore that healing is a divine thing, not arcane- so natural arcane casters cannot heal (Divine Soul is obviously Divine, as it Celestial Warlock).
In terms of the scrolls talk: in my tabletop games we personally have it so anyone can cast from scrolls and that they can involve healing spells, however they are rarer than you find in BG3 and it is more an "locking a spell in ink for later".
Wizards shouldn't be able to do many things and it feels like they have been oddly buffed for the game. Cantrips doing DoT is kinda crazy as well as spells appllying surfaces when they shouldn't. The whole burning effect from fire spells needs to go imo. They also should have to only be able to learn spells from scrolls that are still on their list, pay gold for it and only do it during a long rest (i.e the campfire).
Also switching spells whenever once again is buffing the class a l o t.
