Is it possible that the other party was at the bar first and ordered from there? Then they went to sit at a table to eat? I've been in restaurants where this happens often. People order while enjoying their cocktails and sit when the first course is ready. I haven't noticed when I've been at Applebee's, but I don't really pay attention. In my city you can smoke only at the bar, so I'm guessing people straggle in to meet their group, have a smoke and a drink or two, order, and move to a table to eat.

It's not Applebee's fault that your kids were so hungry and can't "hold their appetites". You're the one in charge of feeding them. If it gets close to meal time, get them some crackers. Keep some in your purse or car.
I realize that you didn't plan for such a wait, but it sounds like you cut things too close.

Might it be possible that the laughing, chatting group at the table seemed to be having a good time and the waitress got caught up in it? They were sending out a positive vibe and the waitress reacted?

As others have pointed out, fun groups tend to tip better than smaller, quieter groups. It pays for the waitress to be more attentive to them. Many waitstaff can size up a party quickly and figure out who will tip best. They can be wrong, of course, but many have some type of radar. Bigger tippers, better service.

Maybe some members of this group are regulars at this Applebee's. Again, the waitstaff knows what kind of tippers these are and will provide stellar service to get the big tip and more repeat service. Maybe a group member is the waitress's dad/uncle/neighbor/boss. Yes, it would be nice for everyone to be treated as a favored relative and receive stellar service. In real life, though, it doesn't always work that way.

You don't mention how old your "young" children are or how they were behaving. Both of these might have a bearing on your service. And asking the waitress to "watch" them is uncalled for. What if another party at Applebee's decided that YOU were receiving preferential treatment because the waitress was watching your "spoiled brats" instead of getting their appetizers? What if one child ran for the door while the other poured milkshake all over a customer in the next booth? Did you give the waitress permission to restrain your children if necessary or to take a fork away from them if needed? Clearly, she did an acceptable job of babysitting because you don't seem to have a problem with her care of your kids. Nice work, not tipping someone who KEPT YOUR KIDS SAFE in your absence. She will remember you next time you show up!

It is possible that Applebee's staff doesn't recognize the people as belonging to this particular group. Maybe staff doesn't find the group to be offensive. Maybe they realize that Applebee's is in business to serve food and not to judge its customers by their political views.

Perhaps you should go to the grocery store and purchase food that you can prepare at home.