Country Bears is also a great place in MK to get out of the heat, sun, and humidity and sit down for a while. But you probably won't get a nap there. CoP, though, is another story--great nap material.
An overlooked indoor attraction at DHS is Vacation Fun--a short movie. You won't sleep--well, at least I didn't, since the movie is adorable--but you'll be out of the sun, sitting, and in air-conditioning.
In Epcot, I always go to the Mexico pavilion when I've had it with the sun/heat/humidity. It's very cool in there and it's fun to browse around. The boat ride is also cool--the temperature, I mean, not the ride itself!
DAK is another story altogether. It's my fave park, but unless you want to subject yourself to insect images (ITtBaB), I can't think of an indoor, cool place where one can rest.
Back to the QS table thing . . . In May my sister and I had mobile-ordered at Woody's and we got there maybe 10 minutes early. There were easily 10 or 15 empty tables in the seating area. My sister was exhausted and there was nowhere, except the ground, to sit in the waiting-to-get-summoned area. We pretty much begged the CM at the front and told her that of course we'd give up the seats/table if necessary and she let us come in and sit down before our order was ready. I think we were in the seats for 5 minutes before we got the notice that our order was ready to pick up.
I'm not sure what advantage it is to WDW to not have sufficient shaded seating areas throughout the parks, but there should be many, many more than there currently are. If they're trying to drive business into the stores, it's not working for me, because if I'm hot, tired, exhausted, and overcome, the very last thing I want to do is purchase something that's not a cold drink, a Dole Whip, or another frozen dessert.