There are still lots of people in the dining room. Getting one's appetite sliced off by a sharp closing time is something that happens throughout the restaurant business. Then the cooks disassemble all their setups, put everything away, and clean up. This irrevocable breakdown also occurs after lunch when the restaurant will not reopen for dinner that day. However, the breakdown between lunch and dinner is less complete.
Chefs remain on duty doing prep work.
A simple and completely acceptable way for a restaurant to serve someone who arrives shortly after the official lunch closing time would be to offer to seat the diner, but advise that there may be certain things that the kitchen might not be able to serve. Some restaurants formalize this practice in the form of a bar menu. The meal would not be as interesting as one served at normal times. One of the bitterest complaints diners voice is that they show up at a restaurant before its closing time, but find it closed. This leaves a big question mark over the integrity of the place.