The divergence time analysis suggests all of their family-level clades split rapidly between 9-14 million years ago. This compares with a Fringillidae-Emberizoidea split of ~21 million years (this is a prior, rather than a result, of their analysis). These are definitely not ancient lineages.
To the sizeable minority who align with the calculations of Bishop Ussher (and others of that time), these lineages certainly are unacceptably ancient!:-O