"Its classical mechanics and electromagnetic expressions are a consequence of quantum mechanics, but the stationary action method helped in the development of quantum mechanics."
That is a direct quote from the article in Wikipedia which refers to Feynman's popular book "The character of the physical law". In that book Feynman DID NOT claim that the theory of Quantum Mechanics implies the PLA for classical mechanics, relativity or EM. The closest statement Feynman wrote in that book is this: "In fact it turns out that in quantum mechanics neither is right in exactly the way I have stated them, but the fact that a minimum principle exists turns out to be a consequence of the fact that on a small scale particles obey quantum mechanics." This is a very different statement and it shows a misunderstanding from the Wikipedia editor (and it seems you too). Here Feynman explicitly claims that the fact that there is a PLA in QM is a consequence of small particles obeying QM , that is , they are equivalent.Same way as the fact that particles obey Newton's Laws imply the existence of a principle of least action in classical mechanics, as formulated originally by Lagrange.
This has nothing to do with Feynman. The deep mystery is always why does nature work the way it does. The QM phase answer provides a deep explanation for why least action occurs at a classical level. I am not sure what your educational background is but QM and Classical are far from equivalent. QM looks like classical under many macro situations.
