I actually think we should just start an old-school crown corp that directly competes with the grocery stores. That's what crown corps used to do -push the private sector to do better through competition aimed at serving an important public need.

The crown corp could sell basic foods; produce, bread, simple meats and dairy products, and at a very low margin. The private grocers would have to compete either by tapping into that mysterious private-sector-efficiency to beat those prices, or via luxury grocery products that draw in customers. The crown corp could either build it's own supply chain, or rely on auctions, as needed.