That part of Matthew is where Jesus is talking EXPLICITLY to and about Peter. This is in contrast, again, to the OT patriarchs whom God made covenants with to prosper their lines and grow them into many nations. There is NO MENTION OF ANYONE BEYOND PETER.
Why? Because, IMO, PETER STILL HAS THE EFFING KEYS! Right now. This very moment. PETER. Not anyone else. Never anyone else, unless God will it in the future.
This also mirrors the passage of prophecy about the Cornerstone Who Was rejected. In my understanding, this is very similar language to that descriptor of the then coming Messiah. It ties in with the language of Jesus being the vine and we are the branches. Peter was another rock in the foundation of the fellowship of believers.
You may say "well, the popes have just been more of those rocks," to which I reply, possibly, except that this is not how God chooses those He calls to sit in authority. Humans don't get to choose those, and, in fact, God loves to choose the least likely, whereas humans often decide on the most likely. I see NOTHING in the new testament that would change this pattern, and since there is a change, that shows me that this is NOT of God, since he is the same yesterday, today, and tomorrow.
So, yes. If Scripture supports me view, then it is correct. I do not see anything other than out of context reading of scripture as a support for the papacy, specifically in this case because it is a simple fact that Peter still has the Keys to the Kingdom, or, will be given them. Not anyone else, especially not someone who is, falsely (IMO) chosen by human agency to wield some kind of authority in a position that was never intended. (I see nothing about a pope or a single leader of the body of the church other than Christ Himself..)