The mind would be perhaps be not quite as blown but the modern large urban area supermarket would be pretty mindblowing to someone from the 1960s. You could probably get a lot of the stuff the random gourmet-ish cook might want but it would probably involve some combination of Saturday farmer's markets--which might have other things like beef--(which my mother did with some regularity), maybe a separate fish market (ditto), a specialty gourmet store, ethnic market, etc.

And, as you say, things like out of season fruits and vegetables or the variety of things like spices and teas would be largely unavailable outside of maybe specialty stores in the largest cities.