Saying that particles explain why electrons all have the same charge presupposes that all particles are identical, but why should that be? Your particle theory has to explain why all particles of a given type have the same property; or rather, why there are types of particles.
The field idea, as discussed here, actually explains why all electrons are identical: because they're all quantized excitations of the same field. It's the quantization that really does the trick, but after all that's how we got started here.
If I understand it a free particle is not really quantised; it can have pretty much any wavelength. But when it is confined by boundary conditions (e.g. within an atom) we then get interference and reinforcement in the wave equations.
Boundary-induced quantisation does explain the limited set of acceptable/stable wavelengths around e.g. an atom.
However, what it doesn't explain, is why the amplitudes of such wavefunctions should be what they are. For example, why should the integral of the squared norm of the wavefunction for an electron be 1, if it is not a particle?
The field idea, as discussed here, actually explains why all electrons are identical: because they're all quantized excitations of the same field. It's the quantization that really does the trick, but after all that's how we got started here.