Dear Ben Baten,
I found your essay extremely interesting and agree with you that the century old assumption of a 'point'-like electron is clearly wrong. You have covered most of the relevant parameters and shown how arguments might be put forward that 'explain' each parameter.
I believe that the 'quantum loop' model is misguided. There is no realistic explanation for why a particle would circle with Compton radius at the speed of light, and it requires a stretch to obtain the known parameters based on this Zitterbewegung model.
My own model assumes an intense C-field -- easily encountered at the big bang or in LHC collisions -- where particles are created. The vortex in the field, combined with the relevant field equations describing the self-interacting behavior of the field, results in an ever-faster spinning, ever-tightening vortex in the same way that a skater spins faster when she pulls her arms in. The significance of the Compton radius is that this is the point where the vortex 'wall' reaches the speed of light.
Instead of assuming that some 'point-like' particle races around a circle with Compton radius at the speed of light, I assume that a new phenomenon occurs here -- the creation of charge (the C-field is uncharged). This adds a new self-repulsion force to the centripetal force that affects the mass of the vortex wall moving in the original field [plus self-induced dipole field], and this force provides the limiting condition that prevents the vortex from following it's natural tendency which is to shrink to an infinitely dense 'point'. Instead, the charged vortex wall resists being compressed to a 'point' and the electron radius stabilizes at radius about 10^-18 meters.
As you point out, it has been known for 80 years that the electron must "spin faster than the speed of light" to account for the magnetic moment but no one has ever explained why this is forbidden. Nowhere in the relevant equations does 'c' show up to limit this action. In fact, if the skater could pull her arms in to a radius approaching zero, she too would spin faster than the speed of light. If angular momentum is conserved, then the speed is determined by the radial arm -- there is no 'natural' limit. It is *NOT* the same as 'boosting' linear velocity to the speed of light.
If one accepts my key assumption, which is that electric charge appears when the vortex wall reaches the speed of light at the Compton radius, then everything else, including the fine structure constant, falls out. And all particles are automatically 'indistinguishable' since there is no possible "marking" that could be seen on a particle spinning faster than light. This stable particle possesses a finite radius, a spinning charge that induces the correct magnetic moment, spin-one half and the particle spin provides the Zitterbewegung frequency. I present this process in great detail in The Chromodynamics War where I also treat all of the other particles of the Standard Model [except the Higgs].
As you know, recent experiments have shown that the de Broglie model of real particle plus real-field-based wave function provides the best explanation of the observed results. To see how such a non-point electron induces a wave function, please read my current essay, The Nature of the Wave Function. I would very much appreciate your comments.
In summary, I agree with you that the electron properties derive from structure and it's high time that physicists focus on this structure. It is inconvenient from the perspective of point-based quantum field theories and QCD, but these theories have a number of problems, not just the "fermion sign problem" that nmann mentioned. I hope that your essay succeeds in focusing attention on this need, and I hope that you give some consideration to a new model that does not involve an electron going in circles at the speed of light.
Good luck in the contest,
Edwin Eugene Klingman