It seems quite a few modern birds (Aves) lineages survived the K-Pg extinction (at least 5, last I checked), but when exactly they diversified is apparently still a contentious issue. The common ancestor almost definitely lived sometime during the cretaceous, so not THAT long ago in the grand scheme of things, but it definitely lived either before or during T-rex’s reign.

I was referring to Avialae, which is the clade defined as all dinosaurs more closely related to budgies than to deinonychus. Many of them would have seemed quite birdy to us, but like the other dinosaurs not many of them made it to the current day and the ones that did are all Aves.

Now my head cannon is that's exactly what happened.