This is yet another example of attempting analysis based on a very questionable primary dataset. "The tooth number and crown size are sampled from the 31 well-preserved 116 specimens of the 21 critical genera of Mesozoic birds." 21 Mesozoic birds with skulls? Is this the year 2002?
Issue #1 is reliance on a topology that's not consensus because there is barely any consensus on topology within Enantiornithes, in non-Aves Euornithes or among the three classic basal avialans (Sapeornis, Jeholornis, Confuciusornis). So yeah, Gobipteryx sister to bohaiornithids, Eoenantiornis sister to that and a Cathayornis plus longipterygid clade sister to that? Just one idea. Bohaiornithids being monophyletic themselves (with that content) is itself disputed. I can't even recall a paper with confuciusornithids sister to omnivoropterygids like this paper's phylogram.
Issue #2 is taxon exclusion. No toothless Zhongjianornis or toothed jinguofortisids? You think instead of having Gobipteryx as the only Late Cretaceous and the only toothless enentiornithine, having Gobipipus, Imparavis, Navaornis and Yuornis might affect results? And in Euornithes, no Meemannavis or Xinghaiornis among toothless taxa, or Iteravis, Gansus, Juehuaornis/Changzuiornis/Dingavis, Shuilingornis, Hongshanornis or Khinganornis among toothed taxa?
"we quantify that the complete tooth loss occurred independently in at
least four lineages (e.g., Confuciusornis, Gobipteryx, Archaeorhynchus,
Schizooura), and the key ancestral nodes are not edentulous." I would say DUH, but I don't think this paper quantified Archaeorhynchus not being a schizoourid at all, and just assumed that based on whatever topology they accepted as correct.
Look, as much as everyone might want higher level analyses, they are worthless until topologies are accepted as consensus. And that requires more and better primary analyses.
Mickey Mortimer