Jesus was not "chosen" to do anything. He was the Word, the Son of God, who purposefully and willfully came to earth, taking on human flesh, as a part of God's eternal plan of redemption through penal substitutionary atonement. His primary purpose was to die for our sins in our place.
I agree with much of what you have said here.
However I believe Jesus is called the Elect One.
So in that sense I believe He was chosen.
I also believe those who are in Christ are the elect, since He Himself is called the Elect One.
“Behold! My Servant whom I uphold,
My Elect One in whom My soul delights!
I have put My Spirit upon Him;
He will bring forth justice to the Gentiles.
He will not cry out, nor raise His voice,
Nor cause His voice to be heard in the street.
A bruised reed He will not break,
And smoking flax He will not quench;
He will bring forth justice for truth.
He will not fail nor be discouraged,
Till He has established justice in the earth;
And the coastlands shall wait for His law.”
Thus says God the LORD,
Who created the heavens and stretched them out,
Who spread forth the earth and that which comes from it,
Who gives breath to the people on it,
And spirit to those who walk on it:
Isaiah 42:1-5
Here we see the Spirit of Christ, referring to the Man Jesus Christ as the Elect One.
So I agree with you that God the Son (The Spirit of Christ) the Eternal Word, was not chosen for anything.
However, the Man Jesus Christ was indeed called the Elect One, and was chosen to bring salvation to the world.
JLB