Before I visited I brushed up a bit by looking some things up and watching some lectured on YouTube. What I discovered about the theme that holds the stories and paintings together surprised me. The frescos all represent Peter copying or emulating the life and philosophy that Jesus taught to his followers.
The frescos are designed to literally and symbolically put our duties and roles as a good human being, in perspective.
(BTW, I was brought up Jewish and I'm an agnostic and I still believe in the messages I saw in these Renaissance frescos.) I guess that makes me a Christian socialist?
Peter was is represented as a model of responsible, kind, civic minded, Christian charity and the frescos all are illustrations of what a good Christian was supposed to do in 15th century Florence. I'm not making this up.
In Florence in the 1400s it was considered a patriotic duty to take care of the poor, and to use excess wealth to not only build hospitals, but to make the city beautiful by using one's wealth to help the city and the less fortunate in emulation of both Peter and Jesus.
One of the scholars I watched, commented that a person looking at the frescos didn't read them in an order like a comic book, but would see them as sermons in paint. Often when visiting the chapel, a priest or monk would act as a guide and point out the various interpretations of charity it showed. He would discuss the meanings and interpretations with the people he was showing around the chapel.
Part of the thing that Lippi, Masolino, and Masaccio did was that the environments, city shapes, and clothing weren't from the 1st century, the city and clothing that the stories were set in were 15th century Florence. This was to make the viewer understand that Peter's acts weren't old fashioned that he was showing you what we need to do right now:
Peter giving money to the poor. Giving away wealth.
Peter healing the sick and crippled literally with the passage of his shadow across their bodies.
All the while acknowledging that God would provide even if you couldn't as he did in the painting by Massacio called "The Tribute Money."
This stuff didn't stop when you left the chapel. In the dining room where the clergy dined, Jesus and the apostles are represented at the "Last Supper" facing you from the other side of the table.
The rich and privileged of Florence in the 1400s knew what Jesus expected of them, why don't the rich and privileged "Christians" today emulate Peter and Jesus?