To me, "riding only" means walking is not allowed at any time, and how many great ones have that rule? Likely not many.
Many of the courses listed in this thread are not carts-required. Kapalua, Tobacco Road, Royal New Kent, Stone Eagle, & Ballyhack all allow walking....I know because I walked them all, except Stone Eagle in the 110+ degree heat (Drop it to 90, and I'd walk it). Quite surprised to see Tobacco Road mentioned at all....there are, what, 3 mildly long walks between holes on the entire course? 9-10, 12-13, and 14-15. Walked it all 8 times I've played it.