Why does the Schwinger limit fail to predict the electron radius? One might think that the minimum electron radius is given by the Schwinger limit, since, were it smaller, spontaneous radiative effects might begin to occur. This value is 3.3*10^-14 m, but we know that rMax is about 10^-22 m. So what do we suppose goes on over those (at least) 6 orders of magnitude where the electric field exceeds the Schwinger limit and yet we see nothing untoward?
 A: Why does the Schwinger limit fail to predict the electron radius?
Because the electron radius isn't <10ˉ²² m. That's a wrong inference. It's quantum field theory, not quantum point-particle theory. The electron's field is what it is. In QFT it's an excitation of the electron field, not some point-particle speck with a field around it.
That's the short answer. To justify it I need to take a detour into fundamental physics. See where the Wikipedia Schwinger limit article says the limit is typically reported as a maximum electric field before nonlinearity for the vacuum of
$E_S = \frac{m_e^2 c^3}{q_e \hbar} \simeq 1.3 \times 10^{18} \, \mathrm{V} / \mathrm{m}$
Well, in classical electrodynamics the electron doesn't actually have an electric field, it has an electromagnetic field, or is such a field. We make it out of electromagnetic waves in pair production, it has a wave nature, we can diffract it, it has a magnetic moment, the Einstein-de Haas effect demonstrates "that spin angular momentum is indeed of the same nature as the angular momentum of rotating bodies as conceived in classical mechanics", the Schrödinger equation is a wave equation, in atomic orbitals electrons "exist as standing waves", and we don't talk of spinors for nothing. IMHO there's good scientific evidence that electron is a field-variation, a wave, in a closed spin ½ path, such that what was a field variation now looks like a standing field. IMHO to visualize that field, you note section 11.10 of Jackson's Classical Electrodynamics where he says "one should properly speak of the electromagnetic field Fμν rather than E or B separately". Then you take a tip from Maxwell who combined curl and convergence like this:
 
...to combine radial electric field lines with concentric magnetic field lines like so: 
 
Note how the electromagnetic field (on the right) is looking something like depictions of the gravito-magnetic field and vector fields. Note too that the electron's Compton wavelength of 2.426 x 10ˉ¹² m is twice the circumference of the inner circle because it's a spin ½ particle with a 720° rotation. But also note that the radius of that circle isn't the radius of the electron, any more than the radius of the eye of the storm is the radius of a cyclone. Talking of which, you can hopefully envisage that electrons and positrons move towards one another and around one another like cyclones and anticyclones: co-rotating vortices repel, counter-rotating vortices attract. When we only see the linear force we talk of an electric field, but there isn't really any such field. The linear and/or rotational motion is the result of two electromagnetic fields interacting, it takes two to tango: 
 
And whilst it exists, positronium doesn't have much of an electromagnetic field because the electron and positron have "exchanged field". This is described as a virtual photon exchange, wherein virtual photons are field quanta, but they aren't actually throwing photons at one another. Ditto for electrons and protons. Hydrogen atoms don't twinkle. 
Anyway, I didn't mean to write so much, but I hope it's enough to justify what I was saying. I hope you can now appreciate that saying the electron has a maximum radius of 10ˉ²² m is like hanging out of helicopter probing a whirlpool with a bargepole. Then when you can't feel the billiard ball, you say the billiard ball must be really small. But there is no billiard ball. Because it's quantum field theory. Everything is field and waves. 
A: My understanding is anything "untoward" is taken into account using renormalization - virtual electron-positron pairs screen the bare charge.
