25 When Peter entered, Cornelius met him, and fell at his feet and [t]worshiped him. 26 But Peter raised him up, saying, “Stand up; I too am just a man.”
t. Acts 10:25 Or prostrated himself in reverence
It isn't even so much that Priests and Popes give themselves the religious title of "Father" its the context in which they do it. They claim to stand between God and man. They claim to be "Alter Christus" and "persona Christi" which to my understanding is in place of Christ and in the person of Christ. Why didn't Peter accept worship as "persona Christi"?
Sorry if you don't like this answer. I will try to put together a more in depth response but it may take a while.
We need to read the passage with St Peter and Cornelius in context. Remember that Cornelius was a pagan Roman before meeting St Peter. He was a man well accustomed to worshipping people (e.g. Emperors) as gods. Note that St Peter doesn't tell Cornelius not to kneel or kiss his feet because that act, in itself, is wrong. He tells Cornelius not to do these things BECAUSE 'I am a man as you are'. The implication here is that Cornelius thinks St Peter is MORE than a man, that he's trying to worship St Peter as a divine being.
The OT is full of instances of people bowing or making other gestures of respect to each other (take Jacob who bowed seven times before Esau his brother). There's no implication of worship in the act itself of kneeling or bowing, it seems. The act of bowing to each other and kneeling before men is not the problem, it is the intent of worship that St Peter corrects.
When we bow to our priests or bishops, we are not worshipping these men. Even when they are celebrating the Sacraments and standing in persona Christi we do not worship them as Christ present. We worship only one Lord, Jesus Christ. We respect the men God gave us to function in the Church and show them due reverence. I can't link directly to the article in the Catechism but if you will follow this link and scroll down to paragraph 1548-1551, you can read a short segment on the Church's teaching regarding in persona Christi Capitis.
I would ask you to go back to my original response to you. Rather than starting over in our conversation, I would like you to deal with the examples of religious fatherhood I gave you from the New Testament. Do you have any issues with those? Let's have a conversation, not just throwing random thoughts, please.