I don't know why Cal-Mart's celery tastes the best to me. Also, they carry (a bit hard to find, near the mushrooms) Japanese cucumbers. Expensive but the only other place in SF I've found them is Japantown and those were older when picked, more seeds, etc. (My favorite place is Monterey Mkt in Berkeley -- they get them in on Wed. but it's a trek to B. plus the bridge toll.)
I walk up to Bryan's for meats.
For what it's worth, there's another Cal-Mart in Calistoga. I assume the Cal-Mart in Laurel village isn't a San Francisco Original.
The OP included some examples like G&G that are mini chains, so I think mainly they're looking for things that aren't Safeway, Albertson's, or Lucky. If that's the case, Andronico's and Mollie Stone's/Tower Market could be included.
Rainbow, Pacific Super, Mission Market Mall, Bryan's, 22nd and irving Market, Cala, Royal Market, Falletti, Ashbury Market, Marina Super, Cal-Mart.
Bi-Rite. Not exactly a supermarket but has nearly the same selection crammed into its small space.
1745 Folsom St, San Francisco, CA 94103
2815 Diamond St, San Francisco, CA
3445 California St, San Francisco, CA
308 Broderick St, San Francisco, CA
22nd & Irving Produce
2101 Irving St, San Francisco, CA
Royal Market & Bakery
5335 Geary Blvd, San Francisco, CA
Mission Market Meat Department
2590 Mission St, San Francisco, CA
2900 Alemany Blvd, San Francisco, CA