It is generally agreed that the commentarial tradition started before the texts were written down. It is well known that Buddhaghosa was just a compiler of an existing tradition that must have existed for centuries before him. In fact the most logical view would seem to be that the commentarial tradition started as soon as the Buddha had spoken a particular discourse. We can see this already in the suttas, wherein some of the monks would give expositions on teachings given by the Buddha. It could even be argued that the Buddha himself sometimes gave commentaries on his own earlier teachings. It would have been a matter of subjectivity whether a particular teaching should be included in the suttas or assigned to a new category of literature called “commentary,” but it seems likely to me that this new category must have existed from a very early point, at the latest from a few decades after the Buddha passed away. And this of course is well within the period of oral transmission.
The commentaries would of course have continued to evolve, as there was no particular reason to restrict the material that could be included in this genre. For this reason there is probably only a subset of the commentarial material that could have influenced the suttas directly.
For Ven. Analayo’s view, have a look at these essays. His most thorough analysis of the relationship between commentaries and suttas is unfortunately not public domain, but for your information the name of the essay is “The Influence of Commentarial Exegesis on the Transmission of Āgama Literature.”
