Has the most recent common ancestor of all living birds lived after dinosaur mass extinction?
I suppose that by the mass extinction, you are referring to the Cretaceous–Paleogene extinction event that happened 66 mya. The Most Recent Common Ancestor (MRCA) of all birds lived about 113.3 mya (early Cretaceous, according to this in oneZoom.org).

So yes, the MRCA of all birds is definitely older than the Cretaceous–Paleogene extinction event. As you said, this also mean that some ancient dinosaurs are/were more related to some extant birds than some extant birds are related.



You (the OP) might not need it but some other readers might need clarification about phylogeny with the post If dinosaurs could have feathers, would they still be reptiles?