The standard answer to this is found by replacing the dynamical sources in the interacting field theory with classical sources, by adding a term $J(x) \phi(x)$ to the otherwise free scalar Lagrangian. The source $J(x)$ models the location of excitations of other quantum fields, in the limit that they are heavy, or moving in a way prescribed externally, and the path integral gives the vacuum-persistence amplitude, which in Minkowski space is $\exp(iEt)$, where E is the vacuum energy, in this case, the energy of the source-vacuum.

The source problem is a simple quadratic path integral, which I write in the Euclidean version:

$$ Z[J] = \int e^{- \int_k (k^2 + m^2)|\phi_k|^2 + J(k)\phi(k) {d^4 k \over (2\pi)^4}} D\phi $$

$$ {Z[J] \over Z[0]} = e^{\int {|J(k)|^2 \over k^2 + m^2} dk} $$

Going back and forth from k-space to x space converts multiplication to convolution, and the free energy log(Z[J]/Z[0]) is:

$$ \log(Z[J]) = \int \int J(x) G(x-y) J(y) d^4x d^4y $$

This path integral is also easily done by summing independent diagrams, where the only diagram describes a particle created by the source, propagating, and annihilating by the source. This instructive exercise is done in an early lecture of Sidney Coleman's Quantum Field Theory course (1975-76 version here: https://www.physics.harvard.edu/events/videos/Phys253 ).

If you want to find the static potential energy between two point sources, substitute $J(x,t) = j(x) = g\delta(x-a) + g\delta(x)$, two static points. To regulate the long-time limit you can consider J(x,t) to turn on and off gradually over the interval [0,T], to regulate the point-source, you can imagine the delta function is fattened up a little. The double time-integral just tells you that the vacuum phase keeps rotating as T gets longer. The energy of the source configuration is the remaining integral:

$$ E[J] = \int J(x) G(x-y,t) J(y) d^3x d^3y dt $$

When you evaluate the delta-functions, you find three contributions. One is equal to the classical self-energy of one delta-function, and is divergent value of G(0,t). The other is the self-energy of the other delta function. The third is the interesting contribution, the part of the energy that depends on the distance $a$ between the two point sources.

$$ E[a] = g^2 \int G(a,t) dt $$

The energy is the integal of the propagator over all time, which is Fourier transforming time at $k_0= 0$. If you Fourier transform space, you just get the usual propagator, except at the special value $k_0=0$, so you reproduce your observation that the $k_0=0$ propagator is: $$ G(k) = {1\over k^2 + m^2}$$ and is the Fourier transform of the Yukawa potential.

You can consider the classical field as forming a cloud of independent scalar particles around the source, and the absorption/emission of the cloud is the potential energy. The scalars wander back and forth in time, but for static situations, you are looking at all contributions, forward, backwards, and spacelike, since you are substituting $k_0 =0$.

Another interesting classical source to consider is the uniformly accelerated delta function. This is treated by Unruh in 1976, and reviewed in "Quantum Fields in Curved Space" by Birrell and Davies. This motion of the source takes a different integral through the Green's function than the constant-motion one for static sources. The evaluation of the behavior here is made easier by considering new particle mode operators for one quadrant. You can also consider oscillating sources as an exercise, etc, to get intuition for the emission statistics in free field theory.

The variations on this toy problem reveals how classical linear field potentials are recovered from quantum field theory. They can be thought of as describing independent particle propagation from sources to sinks, or equivalently (because it is a quadratic path integral) as the (in this case exact) Gaussian approximation of the response of the field to the source. Full quantum field theory is then obtained when you make the physical sources other dynamical fields.

In full quantum field theory, considered in terms of diagrams, you recovering the usual nonrelativistic theory from the ladder approximation, where you consider some particle legs are slowly moving, exchanging particles along horizontal rungs, and the propagation along the rungs pretty much reproduces the static source situation. This is reviewed lots of places in the 1960s literature, but my favorite is the brief telegraphic explanation in Gribov's "The Theory of Complex Angular Momenta".

The limit of potential scattering in full quantum field theory is appropriate when the particles scatter nonrelativistically slowly, as Vladimir Kalitvianski said already. The instantaneous exchange of particles is approximated by the potential energy as a function of distance to a close enough approximation.

The $\psi$ particle, in the final analysis, can be viewed as carrying a cloud of $\phi$ particles around it, all independently propagating. The $k_0=0$ projetion comes from considering slowly moving particles, so that you integrate over all times of emssion and absorption, projecting out to pure spatial momentum. The scattering in terms of Feynman diagrams is a representation of the response of one particle to other particle's clouds. So your intuition is fine, except that the $\phi$ clouds include all virtual momenta, and the lowest order contributions to scattering (rather than self-energy) are those where the $\phi$ links one $\psi$ to another $\psi$.