“Given the Christian view that Jesus is God, is God His own servant?” Sounds just like a Nicodemus type question..”Can a man be born twice?”
No and Yes!...By Jesus serving the Father as a fully actuated sinless man (as well as being God in flesh), he was the perfect example of how men should serve God. Jesus even served his disciples by washing their feet while exhorting them to Love one another. Jesus serves those who love him by washing away their sins and “dropping them into the sea of forgetfullness and remembering those sins no more.” then telling us...”Go and sin no more”. Then Jesus serves us again by promising us the power of the Holy Spirit...so that men could boldly proclaim the gospel to every creature, then baptizing them in the Shekinah glory of God’s spirit. By serving as a man, by giving up his pre-incarnate Glory willingly, he was the ultimate expression of God’s heart extended to fallen humans.
The premise of your question needs to be adjusted...God became a servant and advocate for man, by becoming a perfect man himself,,,since there was no other man that could lift the curse of sin and degredation because all men have sinned”and come short of the glory of God”. There was no animal sacrifice that could fully cleanse men of sin except God’s own sacrifice of his expressed heart to man..that is Jesus Christ. In Genesis we find indications of what God was intending to do, when he at first commanded Abraham to offer up his son Isaac as a sacrifice...later staying his hand when God saw that Abraham was willing to lose all he cherished in trust to God. Abraham I believe began to get the hint when after sacrificing the” provided ram”, Abraham named the place “God the Provider” (Jehovah gyra).
I hate to break it to you, but that last sentence ("go and sin no more") is one of the well known biblical frauds otherwise known as Pericope Adulterae (you are free to look it up rather than take my word for it). The short of it is: he never said it! It never took place. John never wrote it. Some religious nut decided to add it to the Bible 15 centuries later.
But, hey, your feels-good story is well choreographed. I can see why people fall for something like that.