St Peter is perhaps a bit of a thorny question. There is no doubt that he was chosen by Jesus to be the leader of the Church. But beware of the wordplay - he was Simon, but became Peter as a 'rock' - Latin petrus - the rock upon which the Church was to be built.
He certainly showed great human weakness as you have observed. I feel this was a deliberate ploy by Jesus to demonstrate to us all that even the most exalted person is a wretched sinner in need of forgiveness, and when He forgives Peter time after time He is showing that He forgives us all - time after time.
It is also worth recalling that Peter was a man of humble origins. He was a fisherman, with no pretensions to greatness or intellectual ability. Such simplicity was obviously valued by Jesus as an example to us that it is not necessary to have an academic or regal background to achieve salvation or to give service to one's brethren.
His weakness was perhaps liable to let him down in dire circumstances, yet here we see God providing the strength to endure martyrdom. Wouldn't we all like to think we would have the same outcome!
All in all, I see Peter as a great symbol of hope, that we can depend on God to get us to Heaven if we keep asking for forgiveness every time we fall.
With love and prayers,