I'm struggling with the concept of identical particles in QM. Say I have two electrons, with one trapped on my left $\left|\uparrow\right\rangle_1$ and one trapped on my right $\left|\downarrow\right\rangle_2$. The 2-particle wavefunction is then clearly $\left|\uparrow\right\rangle_1 \otimes \left|\downarrow\right\rangle_2$. If an exchange happens, clearly the wavefunction would be $\left|\downarrow\right\rangle_1 \otimes \left|\uparrow\right\rangle_2$, which is totally different and I can detect this exchange by measuring in the appropriate basis. The 2-particle system does not seem to violate any law of QM to me, and I can do all kind of operations on them including entanglement.
Yet, why is it that when I open QM textbook like Griffiths and Sakurai etc. in the Identical Particles chapters, they always say that this is impossible, that in order to construct a many-particle wavefunction, one must know that
God doesn't know which is which, because there is no such thing as "this" electron, or "that" electron
we (can't) follow the trajectory because that would entail a position measurement at each instant of time, which necessarily disturbs the system
I mean, am I tracking the electrons if I trap them on 2 sides of my table? Even if I am tracking the electron, what is the bad thing about it that makes my many-particle state $\left|\uparrow\right\rangle_1 \otimes \left|\downarrow\right\rangle_2$ invalid to QM? What am I misunderstanding?