The point is not on the mathematical formulation (you could easily make quantum systems to have distinguishable particles), but on the physical interpretation of quantum mechanics.
Let's think for a moment what we do operatively to distinguish between to classical particles. First of all, we need to observe the state of each one, and since the state of a classical particle is described by a point of the phase space, we can easily do that just measuring two observables: position and momentum. In addition, we know/postulate that two classical particles cannot occupy the same state (i.e. point of the phase space) at the same time, and each one has to occupy a state. Placing detectors (in principle) at each point of the phase space, we can then attach a label "$1$" to the particle in the state $(x_1,p_1)$ and a label "$2$" to the particle in state $(x_2,p_2)$, both detected at the initial time. Now that we have attached the labels, we can follow their trajectory through time. That trajectory is uniquely identified by the initial condition, and therefore we always keep the two particles distinct (even collisions would not cause problems, because even if the particles then are in the same position, they must have different momenta since they started from different initial conditions and so we can still distinguish them).
Now, are we able to do the same in quantum mechanics? Are we able to attach labels to two identical particles in different states in order to follow them? The answer is, in general, no (because of the nature of quantum measurements). The first difficulty is to measure the state of each quantum particle, since now it is something that we can't measure directly and in general performing measurements modifies the state of the system (that last property will become crucial later). Nevertheless, suppose that we were able to prepare two particles, each one in a defined stationary state $\varrho_1$ and $\varrho_2$. Suppose in addition that each particle has a non-zero probability of being found in every small region (complying with the indeterminacy principle) $\Omega$ of the phase space (there are always states like that in quantum mechanics). Now the crucial problem arises: how can we attach a label to each particle? Suppose that at the initial time we detect the particles in the regions $\Omega_1$ and $\Omega_2$. It may very well happen that $\Omega_1=\Omega_2$! Now, how could we attach a label to each particle? If $\Omega_1=\Omega_2$, it is clearly not possible; if the two regions are different, we could, but then the quantum measurement has changed the state, and forced the particles to be confined to the regions $\Omega_1$ and $\Omega_2$ respectively. So they can't interact anymore, and any eventual coherence between them is completely destroyed. So we managed to distinguish the particles, but at the cost of isolating them from each other, therefore modifying completely the system. That is not the type of distinguishability we sought. And we actually have no other way to distinguish particles, and not because we are not able, but because the quantum theory (of measurements) prevents it.
Therefore it is quite natural, due to this operative indistinguishability, to formulate a theory that postulates quantum identical particles to be indistinguishable. As always, the then experimental success of the predictions of such theory is the best confirmation of it; however in this case the idea is a priori justified (in my opinion) by the above.