Eh. I can see it working.

Humans are social creatures. We like to feel useful and connect with others. In a world with a replicator in every home, dining out is much more about the social experience than the food. Working in a restaurant would be about community and shared interest. People would volunteer to staff them for the same reason people do any form of volunteer work: they enjoy the sense of purpose, skill-sharing, and camaraderie that comes with it. Plus, with replicators making preparation and cleanup trivial, there's a lot less labor associated with food service.

Lastly, consider that post-scarcity dining establishments that would have no tolerance for rude customers. If someone went full Karen on a volunteer, they'd be banned in a hot second. The social dynamics of such an arrangement would entirely favor the staff: if there are no "paying customers" then there's no entitlement to go with it.

I don't find it all that difficult to envision a set of social incentives that would keep restaurants alive.