In section 6.2 in Peskin and Schroeder (P&S) on vertex corrections, it is shown (eq 6.34) that the physical charge of the electron is given by $eF_1(0)$ (as shown by comparing the nonrelativistic scattering amplitude of an electron from a static potential in QED with the corresponding amplitude one gets from nonrelativistic QM). It is then argued that since we know that the physical charge of the electron is $e$, it follows that we must have the exact relation $F_1(0) = 1$ (and, therefore, since we have $F_1(0) = 1$ already to leading order in perturbation theory, there is no radiative correction at any order to the quantity $F_1(0)$).
But here's the problem with that argument, as I see it: indeed, we know that the physical (measured) electric charge of the electron is "$e$", in the sense that it is roughly (in SI units) -1.6$\cdot$ $10^{-19} C$. But the constant "$e$" appearing in QED (and the quantity $eF_1(0)$) is simply a coupling constant coming from the interaction term $-e\bar{\psi}\gamma^{\mu}\psi A_{\mu}$ in the Lagrangian. Hence, why could one not go the other way around and simply equate $eF_1(0)$ to the measured charge of the electron and use it to solve for what the coupling constant "$e$" must be, no matter what the numerical value of $F_1(0)$ happens to be?
This is qualitatively very similar to what happens in classical electrodynamics (and possibly with renormalization in QFT as well, although I haven't got to renormalization yet, so I am not sure), where the physical mass of the electron is a sum of a "bare" mass and a (formally infinite) self interaction contribution. But this is no problem, as it is only the physical mass that is, well, physical, and there was no reason to equate the bare and physical mass in the first place. P&S appear to make a similar mistake when they equate the coupling constant "$e$" with the physical charge of the electron.
Of course I am not claiming that $F_1(0)$ is different from 1, but rather that the argument presented in P&S appears to be flawed. Thus my question is this: could somebody provide a sufficient argument why $F_1(0)=1$ (either by elaborating the argument in P&S to explain why it would somehow work after all, or by presenting some other, independent argument)?
Edit: Perhaps to clarify: the form factors are defined from a sum of vertex correction diagrams (a few shown on page 185 in P&S), and so the fact that the higher order corrections to $F_1(0)$ vanish must be formally due to the form of these extra diagrams, independent of whatever the value of "$e$" is. Hence, you can't just "define" the coupling constant to be equal to the physical charge, so that automatically $F_1(0)=1$; its value is objectively determined by the form of the vertex correction diagrams, not our subjective choice of what we defined "$e$" to be.
