That's IMO pretty narrow in a perverse way. "Can cast all arcane magic" (never mind that the bard's busy healing and the artificer has recently joined in) is a very simple non-detailed concept, and it's something that can apply to all wizards. All wizards are fundamentally the same this way and only differ significantly in the equipment they use (i.e. their spell book)My quibble here is that I see the Wizard as being broader than the Sorcerer. The Wizard basically amounts to a mage with a book who can cast all arcane magic.
Meanwhile there's more variation between sorcerers because two different sorcerers might not have a single spell in common, with a shadow sorcerer possibly playing entirely differently from a storm sorcerer.
And this is why I don't just agree with @DEFCON 1 but I'm not entirely joking when I suggest that the wizard should be a subclass of sorcerer, with the wizard's gimmick being that they can cast all (sorcerous) arcane magic and use Intelligence to do so.