I can't find any commentary around that equates 1 John's statement that Christ came "by water and blood" to refer to his birth. General concensus is it's referring to the manifestation of Jesus at his baptism (water) and his dying on the cross (blood).
This from the People's New Testament Commentary and seems to be the most dominant view:
The water and the blood refer primarily to the baptism that revealed him at the beginning of his earthly ministry and the blood which he shed at its close.
This from Barnes which gives many options for what this could mean, but explicitly eliminates your interpretation:
This does not mean that when he came into the world he was accompanied in some way by water and blood; but the idea is, that the water and the blood were clearly manifest during his appearing on earth, or that they were remarkable testimonials in some way to his character and work.
1 John 5:6 Bible Commentary
You can read through them all if you like; I just went through the commentaries I own as well. They all flatly disagree with what you're trying to make that verse mean.
You quoted the idea.
"This does not mean that when he came into the world he was accompanied in some way by water and blood; but the idea is, that the water and the blood were clearly manifest during his appearing on earth, or that they were remarkable testimonials in some way to his character and work. An ambassador might be said to come with credentials; a warrior might be said to come with the spoils of victory; a prince might be said to "come" with the insignia of royalty; a prophet comes with signs and wonders; and the Lord Jesus might also be said to have come with power to raise the dead, and to heal disease, and to cast out devils; but John here fixes the attention on a fact so impressive and remarkable in his view as to be worthy of special remark, that he "came" by water and blood."
Since Barnes is denying it, it's obviously a belief by some.
Read the credentials about an ambassador, a prince, a prophet. Now apply the same idea to God-with-us; the credentials are His normal birth, came by water and blood.
1 John 4:3 And every spirit that confesseth not that Jesus Christ is
come in the flesh is not of God: and this is that
spirit of antichrist, whereof ye have heard that it should come; and even now already is it in the world.
Same word. Came in the flesh (by water and blood).
Barnes, I guess, gets this (come in the flesh), not sure why he can't get the other.
" That is, this doctrine is essential to the Christian system; and he who does not hold it cannot be regarded either as a Christian, or recognised as a Christian teacher. If he was not a man, then ... A mere show, an appearance assumed, a vision, could not make atonement for sin;"
See? Came by water and blood is the opposite of the "appearance assumed, a vision".
So, yeah, water and blood and cord and afterbirth are all part and parcel of the reality of God-with-us. To deny it, is to deny the "Christian system".