Old-fashioned perturbation theory and contribution from resonances

Question

In his QFT vol. 1 (paragraph 3.5), Weinberg discusses the so-called old-fashioned perturbation theory (OFPT), i.e. the one based on the perturbative expansion of the Hamiltonian. As a result, in this theory time is dedicated, and there is no explicit Lorentz covariance. He claims though that its advantage is that it allows for indicating the parameter space when different intermediate states produce singularities in S-matrix, such that the perturbation theory becomes inapplicable.

As an example, at the beginning of Sec. 14, he discusses the low-energy electron-proton scattering. In Feynman's language of the perturbation theory, the Hydrogen atom pole is invisible independently of the order of the perturbation. However, the OFPT shows the divergence of the perturbative expansion for the CM momentum lower than $|\mathbf q|\simeq m_{e}e^{4}$, which is of the order of the binding energy of the electron in the Hydrogen atom.

My question is why the main feature of OFPT - that time is dedicated - allows us to see these divergencies, whereas Feynman's theory does not, although the second one is just the resummation of the OFPT in the explicitly Lorentz-covariant form (such that virtual particles in the intermediate states are replaced by propagators). I would be grateful for help.