I'm suddenly getting confused on what should be a very simple point. Recall that the $g$-factor of a particle is defined as $$\mu = \frac{ge}{2m} L$$ where $L$ is the spin angular momentum. For any classical system where the charge and mass distributions are identical, $g = 1$. However, one of the great successes of Dirac's theory of the electron was to show that $g = 2$ instead.
The Dirac equation describes spin $1/2$ particles, so it should also apply to the proton. But the proton instead has $g \approx 5.6$. The standard explanation for this is that the proton is a complicated bound state of quarks and gluons, so we shouldn't expect the Dirac equation to apply. But I don't understand why, in detail, it doesn't! More precisely, any argument I can think of that shows $g \neq 2$ for the proton also applies to the electron.
Naively, we can measure the $g$-factor by using light of very low frequencies, with wavelengths much larger than the compositeness scale of the proton. At these distance scales, the proton should look just like a point charge. The only accessible states at these energies are "spin up" and "spin down", just like an electron. From the perspective of early 20th century physics, the electron and proton looked equally pointlike.
So one would naively think that the electron and proton both obey the Dirac equation. But that's not right, because high energy physics can still affect low energy observables. For example, in QFT, we would compute the $g$-factor using the amplitude $\langle p' | j^\mu | p \rangle$ where the external states are proton or electron states. This can be expanded perturbatively, but for the proton there are large loop corrections because the strong coupling is strong at low energies. Meanwhile, the loop corrections are small for the electron because it couples predominantly electromagnetically, and $\alpha_e \approx 1/137$.
This is all standard. My confusion is two-fold. First, at the time of Dirac, we knew of three apparently fundamental particles, the electron, proton, and neutron. If $g \approx 2$ only worked for one of them, how was this a success of the Dirac equation? How was the failure for the other two explained? It couldn't have been that people said the proton and neutron were composite, because we didn't learn that until 50 years later.
Second off, it would seem that $g \approx 2$ for the electron places extremely strong constraints on new physics. For example, if the electron is composite due to a confining force at scales above what we've probed, then this new force almost certainly contributes significantly to $g$, just like the strong force does for the proton. It would seem that this rules out just about all electron compositeness models, but that also seems like too strong of a conclusion to be believed. Does $g \approx 2$ really show this?