What does it mean to say that our church is "apostolic"?
"And what you heard from me through many witnesses entrust to faithful people who will have the ability to teach others as well." (2 Tim. 2:2)Paul is writing to Timothy about succession - passing on what Paul learned to Timothy, then from Timothy to his successors, and so on. (It's worth nothing, also, that Paul mentions passing on what Timothy has heard spoken, not just what he has read in Scripture.)
Our Church is apostolic because it can trace its history back to Christ and His apostles - those He directly taught and mandated to "make disciples of all nations" (Matt 28:19). (Regarding my choice of Bible passage above - if you don't count Paul among those taught and sent out directly by Christ, read Acts 9:5-6 again!)
There have been good successors and bad (with bad successors in the great minority, I might add). There have been good popes and bad (again, bad in the great minority). From the very beginning, the apostles themselves messed things up constantly - and Peter, the leader of the apostles, in particular. If leadership could only pass to someone that was perfect, Christ's church would have ended with Christ. We know from the Gospels that Christ's church was founded in the apostles, that it was founded in imperfect but well-meaning men, and, most importantly, that it was guaranteed by Christ never to fail, despite how the men leading it might fail (Matt 16:18).