Your argument basically boils down to this: “This is a Catholic doctrine. As Protestants, we reject Catholic doctrines. Here’s a Bible verse as justification.” But if existence of a single supporting verse in the Bible is justification, then apostolic succession IS in the bible. In 2 Timothy 2:2, Paul tells Timothy, “[W]hat you have heard from me before many witnesses entrust to faithful men who will be able to teach others also.” Paul tells Timothy, who he has trained as a successor to himself, to likewise train others so that the succession may continue. The verse from 1 Corinthians that the author purports to reject apostolic succession clearly does not even address the issue, much less refute 2 Timothy 2:2.
As the question notes, the doctrine is strongly affirmed in Church history. In fact, it is affirmed before the Bible that Protestants use today is itself affirmed. Apostolic succession was in fact very necessary for the early Church because heretics simply put their own interpretations on Scripture. Only by demonstrating a line of succession could one verify that a speaker was speaking the truth as taught by the apostles.