Pauli exclusion principle comes from the repulsion between electrons?

According to the spin statistics theorem, particles with a half-integer spin are fermions, that is, they obey the Pauli exclusion principle.

But that does not explain why electrons are fermions in the first place.

An electron has a relatively strong magnetic moment. If we have two electrons whose spins are in opposite directions, the electric repulsion between them is partly canceled.

If the distance is just 10^-15 m, then the magnetic attraction between two electrons is still just 10^-20 of the electric repulsion.

The magnetic attraction may be the underlying reason why every quantum state in a stationary atom can accommodate both an electron with spin +1/2 and spin -1/2. The magnetic attraction only works well in a system of two electrons, not three. That is why a state can hold exactly 2 electrons.

The force of magnetic attraction is too weak. Why can two electrons then fit on each quantum state?

But why the Pauli exclusion principle? Consider the particles in a box model of quantum mechanics. If 3 electrons would be in the ground state, then we would have an electron cloud that is denser in the middle in the box, and furthermore, the magnetic attraction does not help in keeping all 3 electrons close to each other.

To achieve a smoother distribution of the electron cloud in the box, one has to populate energy levels above the ground state.

Todo: prove that the minimum energy is obtained with 2 electrons on each level.

If we would have an atomic nucleus that is orbited by negatively charged bosons (hypothetical boson electrons), then all those bosons would fall into the lowest possible orbit? Classically, if we have a positive charge in the middle and compensating negative charges, those negative charges come as close to the positive charge as they can get.

Conclusion: we cannot explain the Pauli exclusion principle in an atom by the electric repulsion alone.