Did it start with Saturnalia?
In the sense that the Christians were willing to put their own holiday up as competition against that one of the pagans, maybe. The early Christians didn't shrink from doing that and would probably consider it weird when modern people say that to do so must necessarily mean that they were
adopting the pagan holiday for their own use.
But the Romans had a hundred holidays a year, so avoiding all of them would be a feat. And then too, there is a basis for dating the Nativity of Christ as was done. It's not entirely arbitrary. The belief of early Christians was that Jesus was crucified on the anniversary of his conception which, if true, would put his birthday in late December. Also, keep in mind that Saturnalia was celebrated over the course of a number of days, not just one.
All of this being considered should leave us thinking the claim that Christmas "started with Saturnalia" is a weak one at best.