Yes, but if you have two 3rd Century sources from very different geographic locations speculating on the date of the conception, it makes it pretty clear that both the Annunciation and Nativity were not celebrated throughout much of the Church at that time. Origen considered the matter to not even be worthy of speculation - certainly not something one would say about about a celebration already being practiced. Indeed, of the 4th Century sources, I am having a hard time for anyone claiming that such things were already believed and practiced. St John Chrysostom is all for separating the celebration of Nativity in Antioch from Jan 7's Theophany celebration to its own feast on the 25th.
The closest we can come to "already believed and practice" was the widespread practice of the Church of celebrating Theophany in early January, which covered a whole range of events in the Lord's life - a sort of jack-of-all feasts before it became (for us) about the Baptism and (for the West) about the Magi. But no one was claiming this date as the actual date for the birth- or for the Baptism or the Wedding at Cana. That the celebration existed is, of course, good enough for you or me, but I doubt Yesh is willing to accept January 7 as being close enough for clerical work. But the Jan 7 celebration combined with pre-Diocletian speculation placing the date of the birth to December 25th might work for the purposes of the OP as placing Christmas as being above pandering to a pagan audience.