Applying your argument verbatim appears to imply the nonexistence of any strong deformation retracts from $A$ onto $M(A)$. So Blum's Medial Axis Theorem, which claims a homotopy-isomorphism $A \simeq M(A)$, appears to not be obtained by a strong deformation retract for pathological $A$ like your example above. But if $A$ has no small-scale variations, then the strong retract is possible, and I'm arguing the max-centre map is the strong retract $A \leadsto M(A)$.

@glS You are correct I wrote composition in the wrong order, thank you for clarifying! I was trying to say $d\pi \circ d(f^*)=df \circ d\pi$, as per the commutative diagram $f \circ \pi = \pi \circ f^*$. But rereading my answer, I dont think I fully understood the question when I wrote it.

@AaronMeyerowitz Your comment is supported by Dirichlet-Hurwitz's theorem that continued fraction expansions are the most efficient approximants with small deminators $ |\xi - p/q|<cq^{-2}$. The PEA obviously involves huge denominator approximants. The EA yields the most efficient small denominator approximants as studied in diophantine approximation.

@Takirion Your above comment to Liviu better illustrates your point and should be included in your question.

It's something about Cauchy sequences in the background euclidean distance which are not Cauchy sequences in the subset metric. In your bean example, the branches accumulate w.r.t. euclidean distance but are not Cauchy in the subset metric (as measured in the medial axis along the branches). This is some kind of metric condition on the inclusion $M(A)\hookrightarrow A$.

The medial axis is like infinite trivalent tree whose branches accumulate to the vertical line segment $(0,t), 0\leq t \leq 1/2$ in the plane. The points $p_n$ belong to the distinct "branches". So you've identified the fact that $m$ is discontinuous whenever the medial axis is not relatively closed in $A$.