This like many teachings of the Church (assumption of Mary, ect.) is something some Pope decided somewhere along the line.
For the bible only people see Mathias in Acts for an example of apostolic succession. Beyond that apostolic succession is solidified in Sacred Tradition from the time of Jesus. I don't know what else to tell you. You can either believe all the evidence around you and become Catholic or stay Catholic or reject it and not believe.
Isn't ironic that today there is Pope that says you do not even have to believe in God to get to heaven yet the Church clings to this apostolic succession?
All the pope is saying is that if a person has little to no knowledge of Jesus or salvation history and if they follow their conscience they to can make it to heaven. I don't see the irony between this and the doctrine of apostolic succession.
A history ignoring some scripture and literal adherence to others seems to be common across all dominations.
In the Catholic faith in order for something to become doctrine or dogma (or even a discipline) it has to jive with Sacred Scripture, Sacred Tradition and Church teachings. For example the pope cannot a proclaim a newly revealed teaching if it first does not jive with scripture an Tradition (capital "T" tradition).
This is why bible only denominations have difficulty standing up theologically. Most only have the 16th century version of the bible and lack the depth in history and teachings from the time of Jesus.
To practically describe this I like to use this analogy:
You have two groups of boys. 22 in each group. All are seniors in high school. Group one has been playing football since little league. Group 2 has never thrown the ball or seen football being played.
You hand both groups the NFL rule book (the bible) and tell them to play the game.
Which group do you think will play the game true to NFL standards?
It obvious that group 1 will. They will because they have received the coaching (Church teachings) and they have the experience (Sacred Tradition).