A while ago, during my introductory physics course, my professor purposefully neglected to tell us about the existence of the anti-electron neutrino in beta decay; he made it an assignment(on our honor) to propose a theory to explain where the missing energy ran off to. Since I was not very experienced at the time, I did not come to the conclusion that the culprit had to be neutral (to conserve charge), but I was aware of basic particle collisions and how scientists smashed particles together in order to observe smaller, more basic ones. Consequently, I came to the conclusion that when the electron broke away from the nucleus, particle fragments (where the electron broke off) were released as well (so, a bit like detritus which would have accounted for the deviation in the missing energy, as a different number of particles could have broken off each time). Now, clearly, I had an advantage over Pauli (since I was aware of particle diversity, and the separable nature of baryons), yet I am still not certain exactly how he was able to narrow his choices down to a single particle (especially since he was decades away from being able to detect such an elusive creature). How would you have come to such a conclusion (if possible, perhaps another response besides Occam's Razor)?
If I were thinking about it from my own perspective, I would see it as the need to balance energy. We already knew from relativity that mass carried an intrinsic energy with it, and so if I measure something and energy is missing, either I (1) didn't account for some lossy mechanism or (2) didn't measure all the energy coming out.
From here, you are stuck with either a particle that carries energy but has no mass (the photon, which I can measure) or something that has mass but has kinetic energy. As you observed, this particle would have to carry no charge because of charge conservation but also because charged massive particles are easy to detect if they exist and don't decay rapidly. Assuming you're hunting for photons at a known wavelength and didn't detect them, the only choice left to you is a massive chargeless particle.
Energy conservation and charge conservation are the two most fundamental laws of physics, and we have never observed them violated in fundamental processes.