[Cogito]

Remember that, assuming circular orbits (which is fine for this discussion, I think), you have to fit satellites on great circle paths next to each other. For a single great circle you can fit a significant number of satellites in a chain, but turns out you don't want too many like that as all your satellites cover the same part of the earth.

All great circles on a sphere will intersect in two places, which assures a collision if you have satellites in them both (without active avoidance). So you have to separate the great circles on to different sized spheres, or add a little eccentricity to make sure the rings don't touch.

There is still a lot of space, but comparing to the size of the surface of the earth is not very informative when dealing with orbital dynamics.

[beerandt]

But you imply those mitigations must be made to correct the "natural," initial orbital state, when in reality, the orbits were never truly circular or co-spherical.

It's not useful to make a simplifying assumption, if the solution to the problem is to reverse the resulting simplifications.

[Cogito]

After I finished the comment I almost went back to modify that intro, but figured the gist of it would be ok.

What I was trying to do was show how we move from the mental model of dots on a balloon (there is so much space on the balloon! the dots are tiny!) to rings around a ball. In order to make sure the dots don't intersect with each other simply spread them out appropriately. In order to make sure the rings don't intersect each other make some of them slightly bigger, or stretch them a bit.

The resulting orbits are still, approximately, great circles on the same sphere - just perturbed a bit.