I also live in a major American city, FabFiveFreddie. I have five DCs, all born in the US and all very used to being out and about with me and en famille in that environment.
Eleven Madison Park would need to offer a heck of a lot of food nuance for that price. Some chef must take himself very seriously indeed to charge that much. There are many absurdities about the American fine dining experience. Would you feel you could send something back in a place like that? There is a whiff of The Emperor's New Clothes about destination dining. Maybe the presence of a four year old should be encouraged.
However, I do not think the resort restaurant the OP and her DH are headed to is quite in that category. How much less than $500 or so for a couple, drinks not included, can a restaurant charge and still claim to be hoity toity enough to warrant the exclusion of a large class of people from its august confines?
My DCs were a very mixed bag, temperament-wise, NNChange. Yet they managed to learn to eat without fuss, to sit on their chairs in restaurants, to use their indoor voices. This is because I taught them to do that, mainly at home but also when we were out. I love the idea that some children are 'naturally compliant'. Maybe some parents make it look easy?
My DCs were not 'naturally compliant'. It took a lot of work to get them to behave themselves, and a lot of practice. You do indeed have to start somewhere.
Maybe when you have five DCs you are a bit more motivated to get everyone to pay attention to you and behave themselves. Certainly it behooves a parent of a larger than average family to make sure the older children are not forced to miss the sort of social opportunities their peers in smaller families experience just because there is a three year old who 'can't be taken anywhere'.
It's perhaps easier when a family is smaller to shrug and say, 'This too shall pass'. If I had indulged that attitude I would have been sidelined for over a decade and my oldest children would never have seen the inside of a restaurant or a museum.
Also, there are many women who have PND or are at risk of PND who would benefit greatly from being able to go out and bring their babies with them and not be given the stink eye by entitled restaurant patrons. It would make the transition from 'human' to mother a good deal easier if their mother persona were to be made welcome.