How important is the Pauli exclusion principle in the distribution of particles on energy levels

It is usually said that the Pauli exclusion principle is the big arbiter of how particles will distribute themselves along energy levels (especially electrons on atomic orbitals), but how accurate is this statement?

It's rather easy to see for an abstract atom, floating in the void in an empty universe. But the distribution is still roughly the same, as far as I know, if an atom is just far enough from other potentials (for instance, in a gas rather than a solid).

But despite that, technically any perturbation coming from whatever distance should be able to split energy levels, as minutely as that may be.

If the Pauli exclusion principle was the sole arbiter of this, shouldn't then most electrons distribute themselves near the ground state, at very close levels due to perturbations at infinity? I assume that the reason for the absence of such a phenomenon is the interaction between the electrons themselves, is this correct?

If there is a perturbation of the state of the electron due to some distant interaction, that same perturbation would apply to all electrons in that state: in other words, you don't get additional states available because of the perturbation - the character of the existing states changes slightly, but no new ones are created.

Simple example: Zeeman splitting. Two possible states with different energies for the electron appear in the presence of an external magnetic field; but if there were two different fields, there would still be just one splitting, due to the energy difference cause by the spin-up vs spin-down relative to the sum magnetic field.

There is a difference between interactions that split up all the possible energy levels (e.g. spin-orbit coupling and the stark effect) and how the electrons are distributed among these energy levels. The interaction between the electrons belonging to the same atom are accounted for by coulomb and exchange integrals and this interaction indeed contributes to the splitting of the degenerate energy levels. However all of the above interactions merely state what energy levels are possible for the electrons belonging to an atom. They say nothing about how the electrons are in fact distributed over these energy levels.

Other perturbations that come from e.g. the electric fields of other ions merely split up or rearrange the order of the energy levels in the same way that the energy levels are split up or shifted by the interactions inherent to the system that I mentioned above.

However now that we know which energy levels are allowed we are also interested in which levels will be occupied by the electrons. That's where the Pauli exclusion principle comes in. The principle simply states that given the energy levels (labeled by quantum numbers like $j, l, s, m_s...$ ) no two electrons shall occupy the same state (i.e. no two electrons will be labeled by the same set of quantum numbers.) This follows from fermi-dirac statistics. So when searching for the electron configuration of the ground state of an atom you simply start placing electrons on the lowest possible energy levels while making sure not to end up with two electrons that have the same quantum numbers. This way you end up with the configuration that has the lowest energy. Other configurations have higher energies and will belong to excited states.

It's very hard to give the full picture without the sometimes crude approximations that I made within this short answer. There is much more behind atomic physics than I am able to describe here. If you are interested in atomic physics I can defenitly recommend the book from Bransden & Joachain .

The confusion here seems to be between energy values and energy states. In general, an energy value can be the same for multiple energy states, e.g. for a non-relativistic hydrogen atom in isotropic and homogeneous space states with all $0\le l<n$ have the same value of energy for given $n\in\mathbb N$.

Note that when one talks about "energy levels", what is actually meant usually is energy state, i.e. one of eigenstates (wavefunctions, roughly speaking) of Hamiltonian operator, not an eigenvalue.

Now taking the above into account, you could see that whatever the multiplicity of eigenvalues, result of Pauli exclusion principle doesn't qualitatively change: the electrons are still one per energy state.