Even if the stores them selves end up having issues, or if most people choose to continue to go to private stores, the presence of a public option with competitive pricing will anchor prices at other stores.
Specifically, I could see it undermining any attempt to implement software based price fixing among the private grocery stores, where they all use a common piece of software to “recommend prices”, with the software set up to increase prices at an even rate among all the clients so that none of them are undercutting each other. It should be illegal, but since technically no one at the companies are communicating about it, it falls in to a legal grey area. I haven’t heard about grocery stores doing this yet, but it’s been well documented in everything from real estate and renting to frozen potatoes.