Representation of indistinguishability in quantum mechanics

It seems to me like this is basically a question of the form

"Why do we use the model that we do for identical particles in quantum mechanics as opposed to some other one?"

to which the best and ultimate reply is

"Because the model we have has never disagreed with experiment,"

but perhaps I can convince you that there are situations in which the commonly used tensor product formalism + symmetrization (or anti-symmetrization) is natural.

Consider a system of two electrons, one trapped in a box on Earth, and the other trapped in a box on Alpha Centauri.

These two particles are certainly identical; all electrons are, but they are distinguishable in the sense that since they are each trapped in a box, we can call the electron in the Earth box electron 1, and we can call the electron in the Alpha Centauri box electron 2, and we won't confuse ourselves. (See What are the differences between indistinguishable and identical? )

I think you'd agree that in this case, it makes complete sense (and is important) to distinguish between the spin states of each electron by writing tensor product states like $|\uparrow\rangle|\downarrow\rangle$, because when one makes a measurement of the spin state of system, one might ask a question like

"after a spin measurement, was the Earth electron in the spin up state or the spin down state?"

So we see that in certain cases, there is in fact a necessity to have a formalizm that distinguishes between identical particles to reproduce the types of individual subsystem measurements that one would like.

Now, if you had two electrons in a single box, then of course one cannot distinguish between the electrons in the same manner, but if the electrons are, for example, assumed to be non-interacting, then the physics of spin measurements in this case is precisely the same, so it makes sense to use the same model. You could, of course, try to concoct another model that reproduces the predictions of the tensor product model, but why would you when we have one that already has spectacular agreement with experiment?