I am looking for a hypernym for all places that provide food, beverages, or both, of any quality or price — such as restaurant, cafe, fast-food and so on.
The name of the industry in the US is “food service” (or “foodservice”); according to Wikipedia the corresponding term in BE is “catering industry”. Within the industry, individual outlets are “food service operations”.
In ordinary speech the most ‘generic’ term is “restaurant”, which embraces both ‘sit-down’ (table service) establishments and fast-food outlets, but tends to exclude institutional operations such as hospital or corporate cafeterias. “Eatery” is sometimes used, but it’s old-fashioned and implies a down-scale establishment; referring to a Michelin-starred restaurant as an “eatery” would be distinctly jocular.
Travel guides, newspapers and the like will often use verbal headings: “Eating”, “Eating Out”, “Dining Out”, and so forth.
In law the category is often referred to as a "retail food establishment", as opposed to a wholesaler or caterer. Hot dog stands and Michelin-rated restaurants are both included.