That age, Discovery Center would be good. The nearby Amarillo Botanic gardens are nice, though might not be much growing outside right now. The adults could go there while the kid and one parent go to the DC just across the parking lot. There's also a playground at the same place, and a pond with ducks to feed. Some nice walks around the ponds/creek there at the medical center.
Never been to the zoo, so no real advice there.
The aviation museum is just getting off the ground, a few nice exhibits, but not a lot, yet. Don't care about the Big Texan or Cadillac Ranch. Sixth Street/Old Route 66 is full of antique shops. Further afield, Square House Museum in Panhandle is good, Lake Meredith has some interesting scenery and hiking. Alibates Flint Quarries are interesting, but probably not for that age. Buffalo Lake NWR is interesting, but PD is obviously much more spectacular. Caprock Canyons is pretty much the same terrain as PD, but much less crowded. It also has free-roaming buffalo, but it's a long drive from Amarillo.
Palo Duro, he will love to throw rocks in the creek. I'd stop by the visitors' center for a literal overview of the park, and a few exhibits. Almost any trail would be great for you, just go a little out and back. and take water. It's usually hotter in the canyon than on top. Just be aware if you're on a bicycles-allowed trail, and keep him safe.
There are a few areas toward the end of the park road where people slide down the lower slopes of the canyon. I think they use cardboard to slide on.
There is a bird watching blind behind the park store, if he has the patience for that.
Check the park web site, there may be some ranger programs or guided hikes.
If you have time/the weather is bad, the Panhandle-Plains Historical Museum is very good. The old cars, dinosaurs, and cowboy and Indian things might catch his interest, as well as the pioneer town. For the adults,
exhibits on the oil business, a large gun collection, and a special exhibit on WWI and the Panhandle.
If you're coming in on 287 from the SE, consider the Charles Goodnight home near Claude. If you are a Lonesome Dove fan, Goodnight was the real Woodrow Call.
Food, in Amarillo: Tyler's BBQ, BlueSky, Tacos Garcia, Coyote Bluff, Kabuki Romanza. In Canyon: Sayakomarn's, Pepitos, Fat Boys, Feldman's. For upscale in Canyon, Imperial Taproom, and we have a very popular coffee shop (Palace) that has won a few national awards.