You have a nice knack of asking just the right questions
In terms of our Evangelical friends, they may rail against 'tradition', yet they tend to be rather keen on their own traditions; it is the same story with infallibility
On St. Peter, we are happy to concede a primacy of honour to the See of Rome as the early Church did, but we think it a shame that some later Popes have taken that to mean a universal governance from Rome; that was not the practice in the early Church.
On the leadership issue we would be at one with St, Ignatius, that where the bishop is, there is the Church. We do not see the need for a single leader. We would interpret the Petrine verses in Matthew as conferring on St. Peter a headship which can only be exercised in brotherly fashion, not a monarchical position; if the Pope in Rome would commit to this, it would be a huge step forward from our point of view; no harm dreaming
Yes, the Blessed St. Peter did deny Our Lord thrice, but in St John we read that Our Lord restored him thrice. I have a great fondness for St. Peter. How like so many of us, impetuous, especially in words, brave, but also fearful; and yet how many of us would be willing to do as he did and lay down his life for the Lord when it mattered?
He seems to me an entirely appropriate leader of the Apostles, and I find I love him dearly, as he has all my faults - and I'd like to have some of his virtues.
In this is love, not that we loved God, but that He loved us and sent His Son to be the propitiation for our sins. (1 John 4:10)