When looking at spatial distances, we 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, which in turn simplifies the whole idea of distances across cosmological scales, which is why it is so commonly used. 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.

> When looking at spatial distances, we usually want to look at spatial coordinates.

That's not the case, because most objects we're measuring distances to are not at the same coordinate time as us. In fact, these objects are generally in the past.

If you're trying to measure the distance to an object that you've observed, that object is, in fact, along your past light cone. The only coordinate-invariant distance that one can actually define, the metric distance, is zero. That means you have to define an alternate type of distance. Some of those choices (like the comoving distance) don't depend on a(t), while some do (like the light travel distance).

> 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.

I prefer Wald, which is much more mathematically rigorous than Carroll. I haven't read Hartle, so I can't comment on it.