When looking at spatial distances, we usually want to look at spatial coordinates. Note that there are no cross terms with a(t) in the metric, so the expansion factor (which is divided out if you look at the definition of comoving distance) can easily be accounted for. No curvature shenanigans necessary. Ofc that also implies looking at a single cosmological time, that was never disputed as it is also literally part of the definition. This in turn simplifies the whole idea of distances across cosmological scales, which is why it is so commonly used. It might depend on the time you choose, with today's proper distance for example being equal to comoving distance if you could freeze the universe. But you can't really do that, so it makes more sense to talk about comoving than proper distances. All of this is explained pretty well on wikipedia, but if you want you can also take a look at Hartle's or Carroll's textbooks on General Relativity.