“I agree with this interpretation. Do you?”
What is it that makes Barnes’ notes authoritative?
Nothing makes Barnes' notes "authoritative". That wasn't my point. It's simply that I agree with the explanation for the passage and I asked if you did as well. I see no contradiction with this interpretation and other Scriptures. No traps set if that is what you may be worrying about - no need for paranoia.
I know that the Catholic Church takes this a step further by stating that the Popes have all been "successors" of Peter and inherit his exclusive authority endowed by Christ. I don't agree with that, but I have no problem at all with the principle role of Peter at the start of the New Testament church.