That's absurd. Anyone can repent for any reason whatsoever, even an atheist. Repentance merely means change of mind (metanoia in Greek). Anyone can regret something and resolve never to do it again. No faith needed there.
Given the flow of the conversation, I took it for granted that "repentance" referred to "repentance to God for sins against God". In that sense, it requires faith.
Of course, in the case of king David faith was the factor for his metanoia, but I am willing to bet there are people of faith who don't repent of everything, even though they have faith.
That must be the case the vast majority of the time. I would imagine that if one's last act is NOT one of repentance that there are some skeletons that have not been dealt with, in all Christian faiths. However, I am not familiar with what level of confessional particularity your priests would require to constitute "coverage". :)
The preist doesn't require anything. Your confession starts with "I confess to God, before you (priest, your witness)..." The only one you are cheating is the priest if you don't tell him everything. He will bless you and ask God to forgive you assuming you told God everything God already knows, and have resolved noever to repeat the same.
As you say, that's hardly the case, so we all tried to cheat God. But in many instances, we really don't see our sin because of arrogance and pride and have "nothing to confess."