Usually others don't do this much even in the rain. Umbrellas quickly and easily become a hazard in a crowded park. Most people use ponchos in the rain and sunscreen and/or hats in the sun. As you will see people do not pay attention to where they are going. Someone will end up walking into your umbrella and injuring themselves.
Even a small one is more than I want to carry. It would be a huge hassle at WDW keeping others from getting poked. (I have experience with strollers, and it is amazing the number of guests who oblivious!)
Another BIG advantage of plastic ponchos is they fit in your pocket, and they protect you better. In WDW, when it rains, it's often QUITE WINDY. An umbrella is nearly useless for WDW rain.
Even in the summer, you aren't really in the sun all day long. You are under cover far more of the day than you are exposed, except maybe if you go to the pool all day. And if you are at teh pool all day, you'd have plenty of opportunity to snag one of the many shaded tables.
At WDW, one dose of a moderate SPF in the morning is basically all we need unless we go to the pool. In which case we apply a second coat. I could also see needing more if you were going to play golf or something. WDW is not like the beach.
I can't imagine trusting a small child with an umbrella at WDW. Wow!
Beyond that, it's not THAT hot at WDW. Humid yes, but not all THAT hot. Once the humity hits 100%, it rains. After the rain, it's usually more mild. But really temps only go into the low 90's for a HIGH. WDW is less than two hours from coast on two sides - so considering how far south it is, it's not all that bad. I think the states a little to the north of FL are hotter, because they don't have the ocean.
There are a number of ways you can further reduce your exposure. Instead of walking between two points, you may be able to take a boat (Epcot, resorts behind Epcot). In MK you can take the train around the perimeter of the park. There are lots of covered walkways, trees, gift shops to cut through. (Plus the gift shops have AC!)
Summer also means plenty of available evening hours. You can further reduce your sun exposure by maximizing after dark park hours.