Is it fair to say that the "enfleshment" of John 1:14 was caused by the Word?
No, it is not.
The word is simply the will, expression, power and outreach of God in His oral decree. The same author John expressly shows the word as a 'which' or a 'what' and not a person who was 'with' God in the beginning.
That which was from the beginning, which we have heard, which we have seen with our own eyes, which we have gazed upon and touched with our own hands—this is the Word of life. 2And this is the life that was revealed; we have seen it and testified to it, and we proclaim to you the eternal life that was with the Father and was revealed to us.
If he meant 'whom' he would have said so. But no the 'whom' is now the one Jesus who is that human expression of God's will and intention. He is the one they can now touch and see because he is the word of God embodied. Certainly, the nature of the 'word' or 'logos' is totally representative of God in every respect - what God says will be, will be. This representation is now the human Jesus who also is the source of life, power and decree, being given this authority within God's sovereignty.
The 'word', like Jesus when he was on earth until his resurrection to immortal life, is incapable of doing anything of itself/himself.
The idea of devising doctrine on some perceived nuance of Greek grammar, when the text is quite explicit (and supported by itself in other passages), is unsound exegesis to say the very least.