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You seem to be confused between the fact that Peter possessed authority and the fact that he did not always exercise his authority well.

No....it is you that are confused. Peter did not possess any more authority than the rest. Those who say he did, must...... because of traditional, religious requirements. Scripture does nothing to give Peter any special authority and I understand your dilemma.

If any Apostle were to be considered more influential with the early church.....a name other than Paul would be questionable. He was given authority to evangelize [Acts 9:15] any....and everyone. Peter was only given the responsibility to the House of Israel [Matthew 10:5-6]. This is why you never see Peter evangelizing the Gentiles. He did not evangelize Cornelius as he (Cornelius) was already a God fearing man [Acts 10:2]. This is also why Peter never goes to Rome.

Peter did not possess any more authority than the rest.

That's not what Jesus said.

If any Apostle were to be considered more influential with the early church.....a name other than Paul would be questionable

Paul himself publicly acknowledged the subordination of his authority to Peter and James.

Influence and authority are two different things - that's why two different words are used to describe them.

Peter was only given the responsibility to the House of Israel

False, as Cornelius shows.

He did not evangelize Cornelius as he (Cornelius) was already a God fearing man

Plenty of people in the Gentile world of the Mediterranean were God-fearers - i.e. they acknowledged the God of Abraham, Isaac and Jacob as the one true God.

But they did not know about His Son, and therefore they needed to be evangelized. Cornelius was just such a man.

This is also why Peter never goes to Rome.

Peter, of course, died in Rome after ministering there. His presence in Rome is attested by his own words.

Peter was only given the responsibility to the House of Israel [Matthew 10:5-6].

That was during Christ's life. The Risen Christ instructed them--all of them--differently at the end of †Matthew's Gospel.

Mat 28:19 Going therefore, teach ye all nations: baptizing them in the name of the Father and of the Son and of the Holy Ghost.
Mat 28:20 Teaching them to observe all things whatsoever I have commanded you. And behold I am with you all days, even to the consummation of the world.