I've been thinking over this question for a while. It's a little difficult to answer because the 'Church Fathers' didn't think of themselves as Fathers. They were for the most part just Christian thinkers, theologians, bishops, etc., who were engaged in thinking and writing about the Christian faith and the business of theology, but because they were 'first' they had a lot of primary thinking to do.

Around John of Damascus (ca 675 - 749) definitely, but probably earlier, you see some shifts going on. All through John's work, 'On the Orthodox Faith', he talks about 'the fathers' or 'the holy fathers' and he treats them as a group, as authoritative, and as having figured out most important theology. That's part of the premise of this work, it's meant to be a compendium that summarises that faith of the Church as expounded by the Fathers. This shift began before JD, but he really crystallizes and demonstrates it, a sense that 'the Fathers' are a block and have come to an end.

Which is why, going backwards, Church Fathers didn't see themselves as doing that or as anything special - it's a view from later-on that treats the Patristic period as something distinct.