> It turns out that there is a rest frame in which this background looks the same in all directions.

I was really surprised when I heard this for the first time a while ago. Because it means that our universe does have a _preferred_ frame.

it's not really preferred in the sense that it's the correct one - just one which has some sentimental meaning to humans' interpretation.

Well, 'preferred' as in 'different from all the others in a way even aliens would recognise' (ie in accordance with https://en.wikipedia.org/wiki/Copernican_principle ).

Aka. a Schelling point.

https://en.wikipedia.org/wiki/Focal_point_(game_theory)

Well, it can serve as a Schelling point. But it's interesting, even without any game theory.

CMB rest frame is important and special unique frame to use as a reference.

Relativity means that there are no special frames where the laws of physics are different.

the CMB is a bit mind bending; it's related to the question "if the universe is expanding, in the past it was smaller and smaller; so where is the center?"

Good read where you'll also learn about the "The Surface of Last Screaming":

https://ned.ipac.caltech.edu/level5/March03/Lineweaver/Linew...

I understand how there can be no center if the universe has positive curvature. But if the universe is saddle-shaped and non-infinite, surely there has to be a center?

I think not; consider pseudosphere, there is no obvious center. Its embedding in 3D space has infinite dimensions but the object itself has finite surface area.

https://en.wikipedia.org/wiki/Pseudosphere

If you mean object with finite dimensions and negative curvature and no center, there seems to be such a thing, see Fig. n066200e:

https://encyclopediaofmath.org/wiki/Negative_curvature,_surf...

EDIT

Rozendorn's idea seems to be still an open question:

https://mathoverflow.net/questions/52851/the-geometry-of-nad...

Rather than think "surface of a balloon inflating", think "surface of an (infinite?) rubber sheet being uniformly stretched". It does intuitively seem like if there is 0 curvature and no "center", then the universe must be infinite though ...