I propose to look at this from the coordinate representation. The Green's function (propagator) may be built from the equation solutions, so the question is in fact about solutions $\phi$ of an "interacting" theory. Let's simplify the theory to a non-relativistic scattering of a scalar particle with mass $m$ from a static potential. The exact solution is not reduced to a free one, but has asymptotics at large distances describing free physical particles with the same mass. Similar property must be fulfilled by the Green's function. So, if one wants to build an "interacting" theory for physical particles, one should guess **the interaction** correctly. It should not modify the particle mass, but should deviate its path from a free one (scatter, in other words). In the occupation number representation it means change of occupation numbers of physical particles. The on-shell renormalization condition is such a subtraction way that removes the self-mass from the initial "interaction". Renormalization (subtraction) "finalizes" our initially wrongly guess of the interaction of physical particles.

The propagator residue is indeed directly connected with the wave function normalization condition in a trivial way.

**UPDATE of 07 august 2021:**

I would like to clarify once and forever the statement "...* I can't see any connection between "$p^2→−m^2$" and "interaction being turned off" *".

Actually there are many (wrong) speculations about "bare particles" parameters, as if they existed indeed, but were non observable due to absence of interactions. In fact we always proceed from the equations of motions (good or bad) of interacting observable particles. That is why we know their physical parameters like mass, charge, spin, etc. However there exist a regime of very high kinetic energy giving the solutions numerically nearly identical to the solutions of free equations. Thus arose a notion of a "free particles". It is not a regime of coupling tending to zero; the coupling may stay the same (permanent for dressed particles or decaying at infinity or at shorter range for scattering processes), but the kinetic energy may prevail effects of those interactions in our numerically acceptable accuracy for this high energy particle (**the interaction effects like radiation or excitations are also governed with some other equations, which have non trivial physical solutions**). So we (may) study a free particle equations with physical mass at different $\vec{p}$. That is why in the propagators stays always the same physical mass.

The situation is slightly more complicated for dressed particles (infra-particles, for example), however in principle it is the same - our accuracy requirements are fulfilled for "free" equation solutions, so we content ourselves with these solutions. But one should never forget - the origin of them is our "full-fledged equations".

You may read my papers on arXiv about dressing examples and integration of short distance "physics".