We are not told why Jesus wept except perhaps a hint in a couple of verses preceding it.
John 11
32 Then when Mary was come where Jesus was, and saw him, she fell down at his feet, saying unto him, Lord, if thou hadst been here, my brother had not died.
33 When Jesus therefore saw her weeping, and the Jews also weeping which came with her, he groaned in the spirit, and was troubled,
I believe it has to do with what you pointed out concerning the word used of His "weeping".
John 11:35 is the only place in the NT where it is used. Literally it means "to make tears" (silently). The others were "wailing".
What you say is what we are taught in Theology 101 (reformed) and OK - it sounds good like it is what would glorify God.
But what of the scripture: e.g.
Genesis 6
5 And GOD saw that the wickedness of man was great in the earth, and that every imagination of the thoughts of his heart was only evil continually.
6 And it repented the LORD that he had made man on the earth, and it grieved him at his heart.
A reformed brother told me that this was an "anthropomorphism".
But that is the explanation of neo-orthodoxy. Don't want to go there.
Either He was grieved or He was not.
We are told in the NT not to cause God grief:
Ephesians 4:
29 Let no corrupt communication proceed out of your mouth, but that which is good to the use of edifying, that it may minister grace unto the hearers.
30 And grieve not the holy Spirit of God, whereby ye are sealed unto the day of redemption.
IMO , in His sovereignty, foreknowledge and the council of His own will, He chose to undergo the grief caused by man because in the end of days He would receive the greatest ultimate glory, love and adoration from His children.
Ecclesiastes 3:1 To every thing there is a season, and a time to every purpose under the heaven:
HankD