[cygx]

But the cosmic microwave background refers to one single event, that has echoes everywhere, right?

An event is a single point in spacetime, whereas photon decoupling happens everywhere, defining a spacelike hypersurface we use for synchronization (in the idealized scenario).

Subsequently, the CMB allows us to single out a particular reference frame (the one where it looks isotropic) and provides a measure of expansion via its redshift/temperature which we can then translate to cosmological time (ie time since the big bang as measured by an observer following the Hubble flow) via our cosmological models.