I've been doing this for more than 30 years. Never seen one design doc. At least not one that was actually used to guide the subsequent implementation.
They both have their place. Ideally the design doc flows out from the proof of concept to document the non-trivial assumptions and decisions. But nothing beats an actual PoC to demonstrate that something will actually work.
I like the phrase “weaponize” to refer to the evolution of a PoC into a workable application.
