Galati in North York north of Finch and Leslie is a fully fledged grocery store owned independently.
There are a large number of Asian supermarkets scattered throughout North York and Scarborough, I don’t know who owns what, some are by big chains (Foody/Tone Tai/Fresh Value), some are not. Obviously avoid T&T.
E1: BlogTO mentioned Karma Coop which is located west of Bathurst, two blocks north of Bloor.