The reasoning behind the warning to only use "all terrain" rather than passenger car or any "high performance" tires was the perceived increased risk of rollover due to the high amount of cornering traction. It's pretty easy to get up on two tires if you take a corner too fast, and sticky tires will hold on pavement until the vehicle flips. Tires like the Firestone Radial ATX the Explorers shipped with, and even the Wildnerness AT had such poor traction (a good "B" traction and acceptable "C" temperature rating), they would tend to simply slide on the pavement, making a rollover less likely.
The 26 psi deal was because the Explorer was originally a truck, but with the Bronco II suspension, it wasn't easy to give it a car-like luxury ride. The quick solution was to just recommend less air in the tires for more cushion.
Unfortunately, the combo of using cheap tires with a low recommended psi made for overheated tires, blowouts, and rollovers.
It's fine to use whatever tires you want, though.