Thanks to all for the input. It has been interesting learning about American restaurant culture. I've been to the States a couple of times, and the only differences I noticed then were that people eat out a lot more than we do, and that waiting for tables at popular restaurants is the norm. This was, of course, before becoming a parent. Now we don't eat out without googling 'child-friendly restaurants' first, or going to our tried and trusted restaurants that we know have designated play areas.
I think people are misunderstanding the question. There are plenty of places like this in America, most near vacation spots. We go to lots in Bethany Beach, DE and Ocean City, MD. And there are a few smaller ones (cornhole and such) where we live on Long Island. You don't leave your kid there during dinner. It's more like we have a 30 minute wait till we can get a table so my son plays while we watch him. Then we all sit down and order our food. My husband or myself will take turns watching him play at the play space till the food comes. Then we all sit and eat together. No one is dropping their kids there, and these aren't fast food places.
Thanks for this. We weren't looking for a place to drop off our son or set him "loose". He's always in eyesight and supervised by us.
For what it's worth, I think whispering canyon cafe in Wilderness Lodge would be a great fit for your family. They do have a very small play table in the waiting area that has Lincoln logs and of course the stick horse races while you eat.
Exactly what I was going to suggest. They also have a pile of toys in the middle of the lobby (surrounded by cushy couches) to play with while waiting for your table. My kids were sad when our table was ready.
Whispering Canyon Cafe sounds like a great option, too. We're not used to waiting for tables, so that's also good to plan for. Thank you. I've gone ahead and made a tentative ADR here just to cover our bases. Chances are we'll probably end up staying at the resort for that particulatar night on our itinerary. We'll see how the trip and our other dining experiences go.
Trail's End, while it sounds wonderful for children, does seem a bit off the beaten track.
OP, though I do not know of any restaurants like this in Disney (it would be awesome, though!), I just wanted to share my two sense. When my DS was 2.5 we took our first trip. We didn't dine out much prior to our trip bc he was pretty much a nightmare in a sit down restaurant unless we had some sort of screen in front of him. I was so nervous for our trip because we'd be eating out every day. We did a lot of character meals (we had the dining plan), and we were amazed at how well he did. He was so intrigued with characters that my husband and I were both able to chew our food while Jt was hot at the same time! (This was the first time this had happened since my oldest was born!)
So for peace of mind you may want to look into character meals like this. Chef Mickey, Akershus, Crystal Palace, and The Disney Jr breakfast all gave the kids a chance to get up and dance or march in a "parade" too while eating. Hopefully these will work for you the way they work for us. (Btw, that was 3 years ago, and we are now able to eat as a family like civilized humans FINALLY! You guys will get there too

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Thanks for this - I was beginning to feel like a bad parent. I thought my child was just a normal, busy boy more interested in physical play than sitting still at a table for a long period of time, but started to doubt myself after reading about how everyone else's kids sat through meals!

We already have an ADR at Crystal Palace - my son loves Winnie the Pooh, so hopefully this will keep him entertained. I didn't know there was a Disney Junior breakfast. My son loves Jake, Doc Macstuffins, etc. I'll look into it. Thank you for the suggestions! And thanks for the encouragement - I look forward to enjoying a meal out in peace again, without having to manage my son's restlessness!
Finally, we also have an ADR for California Grill, as a special treat to ourselves. But don't worry fellow diners, we'll bring out the big guns for this one (iPad time!) so that we can celebrate our first ever trip to Disney with some signature dining!
