In Massachusetts we have a private grocery store chain called "Market Basket". Its cheaper than the rest and spreading. (After the 2 realtive that owned it had a spat, CEO Arthur T. was removed, causing workers to strike.. When your non-union workforce walks off the job for you, you are doing something right as an owner.)
https://en.wikipedia.org/wiki/Market_Basket_protests
Though Trader Joes (Aldi North ) is here, Aldi just started opening up here. Those style markets are affordable, but don't have the selection of the Market Baskets. https://en.wikipedia.org/wiki/Aldi
I wonder if the fact that these are privately held is an advantage.