Tertulian’s early vision of apostolic succession was quite clear:  In his Prescription Against Heretics, he recognized a need for apostolic succession to connect a widely disbursed group of churches.  "Let them produce the original records of their churches; let them unfold the roll of their bishops, running down in due succession from the beginning in such a manner that [that first bishop of theirs] bishop shall be able to show for his ordainer and predecessor some one of the apostles or of apostolic men—a man, moreover, who continued steadfast with the apostles.”

I appreciate your citations from Cardinal Kasper.  His broad concepts of a “collegial membership in a collegium” and “full communion in faith, sacramental life and shared apostolic vision” merge the objectives of “one holy, catholic, and apostolic church” separately stated within the Creed.   These evolved Roman Catholic views sidestep, or eliminate, the original necessity for an individual unbroken chain from the apostles.