The grit only works where it supports the tool. So as long as you rotate frequently it will just hit new 'high' spots while not affecting the low spots at all. By hitting the interior more often than the edge that's where you'll go deepest, it's more statistics than actual aiming for a some high spot or difference. The tool-on-top or mirror-on-top swap can help to adjust if you've gone too deep by favoring supporting the tool at the edge or supporting the tool in the center causing greater amounts of material being taken away at the edge.

Lots of simple processes can cause nearly perfect geometrical shapes to appear: bounce steel blocks on a vibrating drum for a while (and against each other) and you end up with steel balls that rival ball-bearings in roundness (but not in precision, they will be all kinds of sizes).

The extraordinary precision (under 0.1 microns) is only possible because the surface is close to a sphere; when grinding two surfaces together uniformly in all directions, the result tends towards a sphere (or a flat shape, which is a sphere of infinite radius). If you had to make an arbitrary shape, things would be very different.

If there's a bump (a hill) on the surface, it tends to grind out more quickly than the rest because it's jutting out and it's more exposed to pressure from the grit. If there's a hole, it remains untouched while the surface around it is being ground down. This way everything tends towards the ideal shape. By carefully doing a uniform rotation of the pieces during grinding, the resulting shape is symmetrical - spherical or flat.

Other shapes are also used in mirrors (revolution surfaces generated by a parabola, hyperbola or ellipse), but they are all basically just small corrections to a sphere (which is a revolution surface generated by a circle).

You start with very rough grit to go faster in the beginning and remove most material. Then you just use finer and finer grit as the surface becomes more and more smooth.

In the final stages you switch to polishing. It's a different process where the tool is not hard (glass) but soft (pitch), and the grit is replaced with microscopic powders (such as iron oxide or cerium oxide) which work not only through physical mechanisms but also via surface chemistry.

Also during polishing you use testing procedures which can show surface errors as small as 0.02 microns. The testing process is steering your polishing techniques; you apply corrections, or choose different polishing strokes to deal with various surface errors. Whereas grinding is mostly automatic (with a few basic checks here and there), during polishing your brain is in the loop, a lot.

I'm not sure I can explain it fully, but my understanding is that it is related to the probability distribution of the abrasive technique.

The technique involves random rotation and random motion, using lapping blocks that are the same size or smaller than the mirror. The center of the mirror is the most likely to be contacted by any motion, so it is abraded more frequently and becomes the deepest point, while the edges are worn less. The distribution between "less" and "more" is spherical.

I can at least understand the process of grinding or lapping a surface flat. Here are a couple examples:

https://www.youtube.com/watch?v=jx1D0buRCOY

https://www.youtube.com/watch?v=REeGn4hN1Bg

https://www.youtube.com/watch?v=ATG_UYYfx7w

if you rub two things together, there are only two shapes which could possibly remain in constant contact. the first is perfectly flat which is very hard with only two surfaces (easier with three) the second is hemispheres and if you do the easiest thing possible rubbing two things together they will wear one another into matching convex and concave hemispheres. for some optics that is close enough (small diameter long focal length e.g. a 6" F8) otherwise instead of a spherical surface you may need a paraboloid (or other aspheric)in which case you need to preferentially deepen some part of the curve (~70% of the radius) a few millionths of an inch. but when you do it wrong you go back to doing the easy thing and make it spherical then try again... and again ... and again ... (but I'm not bitter). Till you decide it is good enough!

The minimum standard for good enough is a under a quarter wavelength in the short end of the spectrum of interest . this is because an extra quarter down plus the same quarter back up and you are a half wave out with another part of the mirror which when combined by your eye results in destructive interference of both parts of the mirror.

one last point on the final smoothness, the interplay of glass, water, pitch, metal oxide and mechanical force is imperfectly understood. crudely it may be closer to planing than grinding but that does not explain the oxide particles which are found beneath the surface of a figured mirror. another thought is the an atom in the glass is "stretched" up and snaps back into a lower energy configuration which is more atomically flat

I'm not sure in this specific case, but in general, grinding mechanisms to obtain flat planes, angles, and spheres are pretty well-established. For flat planes, the term is "automatic generation of gages" and is a subtle and clever way to grind three faces in pairs. For curves, you'll need a rotor. Interestingly, you can use a fairly coarse system to methodically generate a very precise curved surface.

Depends on what you mean by perfect. its very precise, but I'm sure if you examined the surface with an electron microscope you'd still see imperfections.

Edit: was incorrect about what I said below. Leaving it for posterity

It's important to remember that the final surface - the one that actually reflects photons - is created chemically by releasing a gas inside a vacuum that very evenly coats the surface with reflective metal atoms. I'd wager that process fills in the ~last nanometers~ or so of imperfections. Or at least averages them out enough to not affect the optical performance.