I suspect the typical pattern that independently lead to beaked jaws in so many archosaur lineages may have been something like this: 1. Lizard lips with interlocking teeth > 2. Lizard lips covering jaws with an overbite > 3. Reduced lips with an overbite > 4. No lips with exposed teeth (most of Archosauria) > 5. Exposed teeth with a hard scaly covering on the skull > 6. Teeth with a rostral 'proto-beak' > 7. A reduction in size and/or number of teeth > 8. A toothless beak.
As I mentioned yesterday, there is no evidence that any feathered animal ever had scales on its head (stage 5), and there's fairly good evidence to the contrary in some fully toothed birds like *Eoenantiornis* which seems to have a head completely covered in feathers. So, this pattern may apply to other archosaur lineages, but apparently not to ornithothoracean birds.
(Did I write *Eoalulavis* yesterday? I hope I didn't. Its head is unknown.)