A fundamental problem with the cosmological principle?

Indeed, if spacetime is homogeneous and isotropic it means the spatial sections are maximally symmetric. Homogeneity means there are 3 Killing vector fields, for which you can pick the x, y and z coordinates. But if you choose other coordinates like radial and angular, it is still symmetric. In fact, spacetime is also isotropic which means rotationally invariant also.

Once you take those 6 symmetries there are not many choices in 3D (the spatial sections) for the geometry of those spaces. The choices are flat, spherical or hyperbolic. Whichever coordinate systems one picks in those spaces (and there are various good choices) the space is symmetric the same way. The rest is simply a time dependence which gives us the scale factor.

See Carrol's lecture notes at https://ned.ipac.caltech.edu/level5/March01/Carroll3/Carroll8.html

The coordinate system which takes the time orthogonal to the spatial slices everywhere is then a preferred frame. It is the comoving coordinate system.

Comoving means that if you are at rest in that coordinate system then you are moving along with the large scale flow of matter in the universe. It is in that frame that everything looks homogeneous and isotropic, in the large. In that frame the cosmic microwave background (CMB) looks isotropic.

Like, Carroll states, please note that spacetime is not static, there is a time dependence. Those comoving spatial slices expand and get further away from each other – a galaxy at a certain comoving coordinate with respect to another will get further away from each other because the scale factor a(t) is not constant. Their distance apart will grow proportionately to a(t).

So, yes, there is a preferred frame, it is the one where the symmetries are explicit. Without those symmetries, we would not know how to solve the cosmological equations.

Also note that galaxies and even us can be NOT at rest in the comoving frame. We are not moving exactly along with the flow of the universe. We and our solar system and galaxy etc have peculiar velocities which account for a somewhat higher or lower concentration of mass in our astrophysical near environment. Our peculiar velocity with respect to comoving is about 370 km/s (which includes the solar system around the galaxy, the galaxy and our local cluster – but look the number up, I am going by memory). Hundreds of km/s are not uncommon.

In fact, when we want to see the CMB and measure any anisotropies, we have to first subtract our peculiar velocities as it will have a directional preference.

The notion of symmetries in general relativity is independant of the system of coordinates used, that is why Killing vectors are used. When we say that a spacetime is homogeneous, it means that it has three spacelike Killing vector fields, which will remain true no matter the coordinates used.

There will then of course be a "priviledged" observer for which the metric components will appear to not depend on any spatial coordinates.