Standing Up, what do you think this influence says about the incarnational beliefs of Catholicism and Eastern Orthodoxy?
Like I've explained before, today there is little impact. Probably everyone here would agree that Christ came in the flesh, regardless of PoJ's description of Christ's birth, which was later picked up and shown to mean from Mary's side (east gate).
2000-1500 years ago, that idea to which today's believers all readily assent (came in the flesh) met something very specific. Here's 1 John 5:6a
This is he that came by water and blood,
even Jesus Christ; not by water only, but by water and blood
Water and blood at birth (and at death with the spearing).
On the opposite end of the belief scale was a deceiver, an antichrist. Look at 2 John 1:7
For many deceivers are entered into the world, who confess not that Jesus Christ is come in the flesh. This is a deceiver and an antichrist.
So, long ago the belief was either came in the flesh (water and blood) or no flesh (no water and blood). The water and blood at Christ's birth (and death) meant He was human, as in God-with-us (one of us).
From that initial contrast between Christians and the world, we can trace the two ideas as they played out in the ECFs and councils. For example, we find this
"As we confess the divine birth of the Virgin to be without any childbed, since it came to pass without seed, and as we preach this to the entire flock, so we subject to correction those who through ignorance do anything which is inconsistent therewith. "
http://www.ccel.org/ccel/schaff/npnf214.xiv.iii.lxxx.html
Without childbed would mean without water and blood, but I doubt any of them would embrace the idea that Christ did not have flesh. They just wouldn't be able to give a reasonable (human) explanation for their belief; that is, Christ had flesh, Christ was born, but without water and blood, like maybe from her side, like Eve from Adam.
So, again, today, the PoJ means little to most believers and most believers would quickly agree that Christ "came in the flesh". What that actually meant long ago has long been lost to the trickling sands of time.