That's the default yes, and it works fine if I play the
Forgotten Realms.
But if I want to use D&D to play, say a game in the
Harry Potter universe but during the Dark Ages then magic users are both sorcerers (as magic is apparently genetic) and wizards (as you need to study how to use it for half a decade). And the D&D distinction between "sorcerer" and "wizard" is redundant.
Or if I want to play in a homebrew where gaining arcane magic involves making a
pact with an elemental spirit that inhabits the body of the mage then wizards overlap with warlocks and the distinction there is redundant.
Having a wizard, and sorcerer, and warlock makes assumptions that there are different ways to gain arcane power. Having just a mage suggests there might be one or there might be many, but leaves the specifics up to the DM and the setting.