Could propeller wash provide sufficient lift to take off - even in theory?
Yes.  The wing doesn't care what is causing air to flow past it.  Headwind, propwash, 747 wake, gopher sneezes.
If the airplane is restrained from forward motion, and the propwash over most of the wing is fast enough, let's say more than the airplane's stall speed, then the wing must generate enough lift to support the airplane, causing it to take off.
It's not so much a matter of designing the wing specially, as making the airplane tolerate the many large props needed, and the weight of the engines.  That much power is too much for efficient cruise.  But it's easy to implement with a one pound model airplane.