I agree with the table service being easier with infants because you know once you are called, you are going to have a table ready. Any place is perfectly fine with infants but you might want to not go to the signature restaurants that have dress codes. Adults who go there are usually looking for a nice adult meal. Infants are still welcome, of course, but the atmosphere is generally more upscale.
For counter service, it's more difficult because you really don't want to all have to go through the line to get your food and saving tables is such a hot topic, although I'll set myself up for flames by saying in your case that's exactly what you need to do. A really nice compromise is to get a place outside the counter service place to "set up shop" and then have one adult go inside and get the food and bring it out.
At Epcot, Liberty Tree Inn outside seating area is very rarely crowded so "saving" a table there is easy. Also, if you don't really want burgers and fries, get your food somewhere else to go and take it over there to eat together. We sat next to a party of twelve and each person brought something from a different country and all twelve shared everything.
At Animal Kingdom, the Flame Tree BBQ seating area is rather large and all outdoor so setting the infants up with an adult while the other goes to get the food is easy.
At MGM, try the Flatbread Grill seating at the back end of the park.