I'm A Banana
Potassium-Rich
You have clearly demonstrated that you consciously exercise "DM fiat", the principle that has gone hand-in-hand with role-playing since the original D&D set -- and to which Kamikaze Midget objects, offering a rigid mathematical construct as "the only alternative resolution method".
I find KM's assumptions about the behavior of RPG participants rather disturbing. That is not what I would expect of "adults, ages 10 and up".
DM Fiat is indeed what D&D has used for 90% of its non-combat resolution from Day 1.
It's about time we kill that sacred cow, especially if we want parts other than combat to become a valid way to play D&D (which as the OP points out, we might need to do if the game wants to attract a broader audience).
I don't think the game should become, as TB says, a game of thespian tea parties, but I think the way to make things other than combat as satisfying and entertaining as combat is to reduce the chance for human capriciousness; that is, reduce the DM Fiat involved. Rather than change focus so that it was game about tea parties, the focus would broaden so that tea parties and butt-kicking medieval mercenaries can go alongside one another: this broadens the player base so that people who enjoy both can both enjoy D&D.
Reducing the DM Fiat involves rules. Rules will involve some math. Like most of D&D, making that math simple and significant (so it's not a long calculation, and so it's not fiddly little bits) is part and parcel of making the game more accessible as well.
I'm sure you could play D&D without any rules, with only DM Fiat, but such a D&D would depend heavily on the quality of the DM. Rather than require a stellar DM, which isn't a common thing in the world, I would do the same thing combat does: put it in a neutral third party, the rules. A stellar DM can always ignore the rules, and will make the game better for it, but more average or slightly-less-than-perfect DMs can use the rules so that they can play to their actual strengths as DMs.
DM Fiat should not be, in my mind, the way to resolve non-combat encounters, any more than it should be the way to resolve combat encounters (and hasn't been, since Day 1).