Generally, if the question is coherent, but slightly confused, it's worth answering straight up. Please don't spend time challenging the frame of the question in answers.
If you do have questions about the intent of the author, ask them in the comments, but don't get into long drawn out discussions (if you feel like it's going to be long, take it to chat).
As far as this question in particular, while the question is not a great one. It's still a legitimate one that can be answered.
I find your complaints about the question to be overwrought though. The question is asking how the Catholic Church explains the fact that Jesus chose an apparently faulty instrument for his work. The question then becomes why is this important isn't everyone broken? That's something to ask the OP in a comment. Maybe they'll clarify.
The title's presentation is a bit off. But only in that they are trying to cram too much into one sentence. It's fundamentally 2 parts:
- The Catholic church believes God chose Peter on which to build his church (and that the office should remain in perpetuity)
- It's clear from scripture that Peter is faulty.
The question asks for a Catholic reconciliation of the two ideas. It does not explain why there is an inherent or apparent conflict between the two ideas. And that's what I feel is missing here. Why are those two things in conflict (is it an erroneous understanding of the Papacy? Is it something else?).
A good answer here explains both why Peter was chosen, why God choses to use flawed vessels, and finally, addresses why there is no conflict between these two ideas. And does so in a presentation that presents sources that a Catholic driving by would consider authoritative (likely a concert of Church writings and their scriptural sources where applicable).