I think you're going to get as many answers as there are restaurants at Disney resorts.
For simple table service dinners, not "signature" meals, specifically in the resorts...
I'd say take a look at the menus, first and foremost, because what one person considers "simple" may not be what someone else does.
That said...
You're staying at Polynesian, so monorail resort restaurants might be easiest. (Unless you have a rental car.) Kona Cafe might be worth trying for dinner. We had a stellar meal there a few years back and a good (but not AS good) meal there on our last trip. We're going back again on our trip this fall. There are interesting choices, but the flavors aren't "out there" for those who want simple, and the kids menu had your basic kid stuff. Easy peasy.
The Wave... at the Contemporary gets great reviews as a hidden gem. It's on my to-try list.
Grand Floridian Cafe at Grand Floridian. We ate here twice on a previous trip. Once for lunch - loved it. Once for dinner - didn't love it as much, but it was still enjoyable, so we've booked it yet again for our fall trip. American food, pretty surroundings, lovely resort. Seems simple to me.
Also at Grand Floridian, if you want a character meal, there's the dinner buffet at 1900 Park Fare. It's buffet food, so of course, there's something for every taste, and there's always fruit, chicken fingers, fries, etc for the kids (and adults, if they wish). The characters at dinner are usually Cinderella, Prince Charming, Lady Tremaine, Anastasia and Drizella. The last three are a real riot, and of course Cinderella is a big draw for the princess-loving crowd. We had good character interactions there.
Off the monorail, but still in the Magic Kingdom resort area, lots of people love the meals at Whispering Canyon Cafe at Wilderness Lodge. It's a gimmicky place where the waitstaff are purposely "rude" (in a joking manner) and my husband does not appreciate that sort of humor, so we've always skipped it. But I know a lot of people who enjoy the silliness and the food there. American stuff with a lean towards barbecue. And again, a simple kids menu, nothing fancy.
If you don't mind going further afield, the dinner buffet at Boma at Animal Kingdom Lodge is terrific. There are some unusual offerings there if you feel a bit adventurous and want to try something new, but lots of American-palate-pleasers as well. If you do this, I'd make an evening out of it - head to the Lodge a bit early, walk out onto Arusha Rock and look at the animals, maybe try to get the night-vision goggles for a little bit after your meal and see if you can spy the animals on the savannah at night. When is your trip? If you're going around Christmas time, the tree in the Lodge is spectacular, and very different from your typical tree. So lovely. Great photo backdrop - and in the evenings a Photopass photographer is there to take photos of families together. A really nice touch, I think. (But totally moot if you're going at a different time of year.)
I've heard good things about Captain's Grille at Yacht Club, but haven't eaten there. My little guy likes to eat, so we go ahead and splurge on the "signature" restaurants - we typically go to Yachtsman Steakhouse when we head over the the Yacht Club for a meal. But I hear that Captain's Grille is tasty, and obviously a quicker, cheaper meal than the fancier steakhouse.
If you can get a reservation, you might also want to try dinner at Beaches & Cream at Beach Club. It's a fairly basic menu, but that's just fine when you want a basic meal - the burgers were delish when we last went. Can't go wrong. Honestly, we'd probably skip the ice cream there and head over to the Boardwalk for the new Ample Hills Creamery for dessert. But if you want a sundae to round our your meal, Beaches & Cream does make good sundaes, too.
I hope some of this helps you make a decision... but again, I definitely suggest looking at the menus for yourself (and understanding that some menus change a bit with the seasons). It's your vacation - go with what you like best!
