Is anyone aware of other games where you improve your skills through doing, with you starting as a vanilla no-class character, as you do in DS1?

Oblivion and Wizardry 8 also have you increasing skills by using them.

final fantasy 2

while not technically inaccurate, phrasing it like that really gives the wrong impression. you start with a class, that you can pick, or you can create one entirely for yourself. the class does not 'enable' anything, it gives starting bonuses to stats and skills, and occasionally an extra starting spell or 2, which are just spells you can get easily enough by playing any class anyway.

Dungeon Master.