You’re not putting a municipal grocery store on the same block as an existing big box and saying “wow, savings!”
Food deserts exist in NYC, and many New Yorkers buy staples at corner stores that charge significantly more than a standard grocery store. Your second paragraph implies that this policy is due to some dislike of existing grocery stores, but that assumes these communities are actually being currently served by grocers at all
Those corner stores exist in a perfectly competitive market, they have low turnover so they have to charger higher prices to pay staff wages and rent. There are no abnormal profits. Why would you set up a government run store with more operational inefficiencies and less ability to respond to the local knowledge problem, than like give money or food stamps to people in these areas, or use the money in some other way?
There is no such thing as a perfectly competitive market. I don't understand why people on HN are obsessed with this term. It strictly cannot exist in the real world.
A perfectly competitive market requires the following (this is not everything, just some):
Food deserts exist in NYC, and many New Yorkers buy staples at corner stores that charge significantly more than a standard grocery store. Your second paragraph implies that this policy is due to some dislike of existing grocery stores, but that assumes these communities are actually being currently served by grocers at all