That's correct. Most translations don't make this intuitive because many try to deify Yeshua, but aside from all of the proofs you gave, the gospels bend over backwards to make it clear that he was begotten. He had a beginning point, has a birthday, etc.
You can also use Hebrews 1:1,2 to supplement your point as well, but you have to make sure you use the right version since most of them are deliberately altered to say God made the world or universe through Christ. That's wrong, God made the ages through Christ.
Hebrews 1
(REV)
1 God, having spoken from old time to the fathers through the prophets in many parts and in many ways,
2 has at the end of these days spoken to us by
his Son, whom he appointed heir of all things, through whom also he has given form to the ages,
It refers to creating the church age through Christ. As far as I can tell, that's the only sense Christ was instrumental in anything that was created.
p.s. - I included a link to the REV Bible. Outstanding version. It was translated by Unitarians unlike most Bibles that were translated by Trinitarians.