For fun, let's take a break and look at a different aspect of this subject. Reconciling molecular studies to the fossil record.
"The early evolution of living birds has been sharply debated, with two disparate interpretations. Molecular-clock studies consistently date the emergence of modern bird orders at ∼100 million years ago or older, coincidental with major continental breakup. This is supported by some biogeographers who use phylogenetics, accept an ancient evolutionary origin and use historical geology to guide their reasoning. The fossil record, however, provides evidence that modern birds represent an explosive Tertiary radiation, following the Cretaceous–Tertiary cataclysm, and their origins are almost 50 million years younger than that predicted by molecular studies. Here, I argue that this explosive, punctuated model conforms to the typical pattern of vertebrate evolution characterized by rapid diversification following a major extinction event."
So in current evolution theorizing about birds there are divergent opinions. How can these be reconciled?
The simple answer is that modern birds developed from pterosaurs. There was no "explosive Tertiary radiation" out of the blue. Modern birds simply developed from advanced pterosaurs - from the pterodactyls.
The molecular studies are pointing to the pterosaurs (the earliest birds); and the recent fossil record is pointing to modern birds.
When pterosaur (early bird) fossils are included in the fossil record, there is a continuous series of bird fossils.
There is no contradiction.