As you know, before Wilsonian POV on QFT, there was an "old-fashioned" renormalization prescription, which was justified/substantiated with different truthful reasonings.
When I was young, the following reasoning looked quite convincing, if not perfect, to me. Let us consider, for example, scattering a low-frequency EM wave from a free electron. QED must give the usual non-relativistic Thomson cross section for this scattering process, and indeed, QED gives it in the first approximation. However, in higher orders the charge acquires perturbative corrections, so that it is the initial charge plus all perturbative corrections who serve now as a physical charge in QED. (Let's not discuss their cut-off dependence.) Therefore, there is no problem at all since we just must define the physical charge as this sum. S. Weinberg, arguing with Dirac, writes in his "Dreams" (page 116), that it is just a matter of definition of physical constants.
We can still find this way of presenting things in many textbooks today.
Isn't it sufficiently convincing for QED? Do you see any loophole in this "old-fashion" reasoning?