Clock synchronization in Kerr spacetime

Is there an intuition or explanation as to why stationary spacetimes like the Kerr metric do not admit good clock synchronization? I am not asking for examples that prove this is not possible, but rather explanations on why it cannot be done, and the obstructions to this procedures in concrete terms.

From what I read/understand, static spacetimes pose no problem because the time-translation symmetry and time-reversal takes care of it. What is not obvious is that why time-reversal is essential: after all, Kerr admits globally timelike Killing vector on the entire exterior (outside ergoregion).

There are also a few other subtle points: (here we assume that the spacetime regions of interest is simply-connected, otherwise there can be extra complications related to time machines or wormholes)

1. Strictly speaking, clock synchronization is a coordinate-dependent thing: clearly, if I am in rotating frame, even in Minkowski space I will mess with synchronization of clocks. In other words, I should be able to phrase clock synchronization as the existence of a (global) chart such that clock synchronization is possible (the inertial coordinates $(t,x,y,z)$ is one of them). What’s the precise definition of this synchronization in geometric terms?
2. What’s the implication of being unable to synchronize clocks in stationary but non-static spacetimes? For example, the Earth is for all practical purposes a slowly rotating geometry, and rotation has visible effects (on our GPS for example). In this case, what does the phrase "clock synchronization" mean physically? For example, if I fly from country $A$ to another country $B$ at different time zones, then back, is synchronization the thing that’s responsible for matching my (GPS-stamped) clock to the local clock at $A$ or is it something else?

In fact, I would appreciate any formal and operational definition of clock synchronization in GR, and what the obstructions are (if any) and how to get around it if it is impossible to synchronize clocks in the sense of special relativity.