It's not just restaurants either; 'place' can refer to almost anything, and works providing the speaker and listener understand the context.

'Place' to mean restaurant has no suggestion of size. You could say 'that big Italian place' and there is no contradiction.

The answer is yes. Just like others have mentioned, one of the most common ways of using the word "place" informally is "my place" "your place" "our place" their place" it just means home, apartment or flat in British English the place where you or others live.

It can be used for a restaurant of any size. "Place" really just means "place," and it is understood to meant "restaurant" from context. It's short for "place to eat." It is more general and less formal than "restaurant." Dunkin' Donuts is "a doughnut place," but you would never call it "a doughnut restaurant!" You could say: "Is there a Chinese place around here?" "Sure, the Kowloon is famous. It's the biggest restaurant in the state."