D.M.
S.,
I agree that ignoring it at this age isn't teaching your son anything and could just be perpetuating the problem. With my son for eating out, I have a whole game plan where we take two cars or if we are meeting someone ask them to meet us there and secure seating and maybe order the food in advance. If you think about it, toddlers have a bandwith of sitting for max 20 (maybe 30) minutes to focus on tasks like eating, coloring, object play and things of the sort. You use your entire 20 minutes just sitting down and ordering food, forget about the wait for food. So, once the toddler is done with his food there is no bandwith left for being patient and waiting for others to finish.
With my family we send the table person, who gets the seats and orders for me and my son and whoever else wants to eat with us. By the time we get there food is on the table and we're able to enjoy the time together, and I usually offer him dessert so it's an incentive.
I always leave in my own car if my son acts out and gets fussy, which can happen if we make plans to eat out after he's reached his peak of activity. So, I plan outings around naptime and when he will be his most perky and awake. It seems like a lot of planning, but it's worked for us.
Also, at home I don't stress about eating at the table or if he finishes dinner. I just love it if he sits with us. Otherwise, if he sits at his coloring table or on my lap that works. But, we always sit down and eat, and my son sits with us as long as he can stand it.
It's really not cause and effect in my opinion, he's done and he has needs that he wants to have met. Toddlers at this stage don't have the patience to sit and wait like adults do. But, you can teach him to sit gradually and develop that kind of patience over time.
Good Luck.