So, in game, why isn't Cure Wounds etc on the wizard list?
Because wizards don't get all arcane spells. There are bard, sorcerer, and warlock spells that are not on the wizard spell list and each of those classes casts arcane spells.
The sidebar regarding the weave in the PHB spellcasting section sums it up. Arcane casters pluck directly at the weave and divine casters access the weave through a divine source that mediates the access. Spells are not listed as arcane or divine; it's the method of accessing them that makes the determination and each class has it's class list of spells. The separation is:
Arcane: Arcane trickster, bard, eldritch knight, sorcerer, warlock, wizard
Divine: Cleric, druid, paladin, ranger
I think it's interesting to note that monks don't appear to access the magic of ki through either method so could be considered neither arcane nor divine, but the description seems to lean more towards arcane magic if a system were to force one or the other. It's also interesting to note that warlocks access the weave directly through magic learned via patron and not empowered by the patron (arcane vs divine distinction) like a cleric does. In theory, a cleric can lose that mediated access to the weave but a warlock would not if it were to become relevant.
Whether cure wounds appears on the wizard spell list or not is irrelevant to whether cure wounds is divine or arcane.