I very much like the first answer, especially the superb visuals!
However, I will nonetheless add a simpler approach to answering your question, which is this: Pauli exclusion holds as long as there is something in space or in spin (spin being the pairs of electrons) that firmly distinguishes two "stable" states.
What that means is that your initial intuition is essentially correct if all the electron pairs shared exactly the same molecular orbital.
So the answer to your question in general terms is that electron pairs settle into molecule-sized orbitals (molecular orbitals) that are physically spread out over the entire ring, but which nonetheless have unique geometric features that keep them distinguished in real space. I would add that they are even more unique when represented in an abstract space called momentum space, where the geometric differences translate into actual distances between the electron pairs. These abstract distances in momentum space satisfy the Pauli exclusion principle just as well as physical separation in real space, so that even very similar looking orbitals can remain distinct under Pauli exclusion.
For more details and some easy visual analogies, you can read more in the addendum below.
The same general problem of how to distinguish electron pairs to meet Pauli exclusion exists in metals. In that case, the conduction electrons are free to drift over the entire macroscopic metal. The result is an even more extreme need than in benzene for the pairs to distinguish themselves firmly from each other, since for conduction electrons all of these gazillions [technical term] of pairs must find "unique places" in which they can maintain their unique identity. That's tough!
Now it turns out that these two problems of benzene electrons and metallic conduction electron pairs are related in a rather deep fashion, since a benzene ring can be interpreted (quite correctly) as a special tiny case of a one-dimensional metal. That means that if you understand what happens conceptually in metals, you are well on the way to understanding all forms of hybridization in organic molecules. So, let's take a quick look at how that works.
The situation for electron pairs in metals is most easily understood by imagining the pairs moving back and forth along the $x$ length of a very narrow, highly conductive, and unusually regular metallic wire. Next, borrow the $y$ and $z$ plane to represent the complex plane (complex phase of a wave function). The amplitude for finding an electron from the pair becomes the radius of the complex plain at each point along $x$. For stable electron pair states, these radii and their phases will exhibit a lot of continuity as you move their as along the $x$ length of the wire. They will also rotate at the same speed in the complex plane of $y$ and $z$.
In fact... if you calculate the stable or "stationary" states possible in such a configuration, you will find that they bear a striking resemblance both visually and mathematically to skip rope solutions, that is, to the ways that you can twirl a skip rope so that it maintains stable, unmoving (the "stationary" part) loops.
So let's look at that situation as a good analog solution to the quantum equations. Anyone can make one large loop, and that in fact represents the electron pair state with the lowest possible energy along the length of the wire, or the base state. Incidentally, the volume enclosed by the skip rope is proportional to the square of the radius, which as I mentioned earlier is the "amplitude" of the wave function. So... that means that if you simply look at the volume enclosed by the skip rope as it turns, you pretty much instantly know what the probability of finding an electron from the pair is. In the case of the base state, it means that the electron pair will tend to be found in the center of the wire, and almost never at the ends.
In chemistry such varying probabilities of finding electrons are usually represented as densities in ordinary space, without the benefit of the dynamics of the complex plane. And for good reason, since for anything except wires, trying to show the complex plane would result in five-dimensional spaces. Those are tough to visualize! Instead, the idea of a phase is attached to tell the relative positions of the rotating points in the unseen complex plane.
The skip rope in the one-dimensional wire makes the idea of phase a lot easier to visualized. For example, good skip-ropers can easily create double loop skip ropes. These correspond to the first excited state of the electron pair, one with a bit more energy added (you have to twirl the rope faster) and two distinct loops. Phase just means this for such a double loop: When one loop is towards you, the other is opposite of you. You can label one such loop positive and one negative, but of course it's all arbitrary, and really just a technique for roughly describing those invisible but rope-like complex plane dynamics.
Notice that the volume-equals-probability rule still applies to the double loop. That means that in the first excited state of the electron pair, the odd are good and roughly equal for finding it in the left loop or right loop, but close to zero at the ends and at the center. So, this really is a unique state for the excited electron pair, one that is quite distinct from the first single-loop or base state.
Now there's an interesting bit of mathematics, Fourier transforms, in which any very long sinusoidal shape can always be distinguished very sharply from any other sinusoidal, even if they only differ slightly in wavelength. People use this idea all the time, since in the case of radio waves (which are sinusoidal waves, mostly) it's called tuning a radio!
The uniqueness of long sinusoidal waves is relevant here because it turns out that Pauli exclusion is met equally well if two pairs of electrons are separated from each other in frequency space.
So, another way of saying all of this is that when electrons are "delocalized" either in bonds or in metals -- it really is the same idea -- they can achieve uniqueness by bumping around either in ordinary space or in a more abstract frequency space called momentum space. Both work equally well. The "extra quantum number" you speculated about in your question could be equated to this idea of additional separation in momentum space for example.
So, the benzene orbitals maintain Pauli exclusion uniqueness by finding specific stable solutions that are similar but geometrically distinct, and which are even more unique when interpreted in terms of frequencies or loop counts. Those equate to a quite different-looking, distance-like separations in the abstract space called momentum space.
So to finish up, what about that extreme case I mentioned of metals that have an electron or two for every atom in what may be a very large crystal?
That's a truly fascinating case, a sort of "benzene gone bad" scenario in which the need for Pauli exclusion uniqueness pushes the electron pairs into absolutely bizarre behaviors, ones very important to modern (and ancient) technologies: Some of the electrons are forced to get very hot. Now that's weird!
What happens is easy to see with the above background. The first pair of electrons in a silver wire is very, very cold, near absolute zero, moving so slowly that it forms only a single loop in the center of the wire. To maintain Pauli exclusion -- the momentum space version of it to be precise -- the next pair must look like a skip rope with two loops, and the third three, and so forth. You get the picture.
But each such loop has more energy, even if only by a little bit! So what happens when you have a gazillion such electron pairs all vying for uniqueness?
Someone gets pushed very, very high up in frequency, and thus also in energy, while other lurk in the deeper and colder regions of this pile of moving electrons. The result is almost like a sea in momentum space, and that is in fact what it is called: A Fermi sea, with cold electron pairs at the bottom, and some truly hot ones at the surface. Fermi seas are also referred to by the more common phrase of "conduction bands," meaning the range of energies that the electrons occupy in order to maintain Pauli uniqueness.
You see Fermi surfaces all the time, since it is only the hottest electrons in a metal that interact with photons and bounce them back at you. You are in effect looking directly at a hot Fermi surface every time you look into a silver or aluminum backed mirror.
How hot? Well, believe it or not, X-ray hot. A typical piece of ordinary shiny metal, say a fork or spoon, would kill you from X-ray exposure if all of its hot Fermi surface and near-surface electrons could suddenly fall back into a "ground state" of just one loop. They are tamed and prevented from doing so by all those other cooler electrons that refuse to let them fall to the bottom of the Fermi sea.
And with that I'll close. I realize the parts about metals are a bit remote from the original chemistry of benzene rings. But then again, as you get into longer and longer carbon chains such as the ones that carry off electrons from chlorophyll molecules, you may find that recognizing the "almost metallic" nature of multiple pairs and the way they interact in momentum space can be a good shorthand for understanding how some pretty complicated forms of large-molecule dynamics play out.