The early Christian appeal to succession was not a vague boast about institutional age. Clement of Rome describes the apostles appointing ministers and providing for approved successors (1 Clement 42–44). Irenaeus then deploys the public succession and teaching of apostolic churches against Gnostic groups that claimed hidden revelation (Against Heresies 3.3). Tertullian makes a similar challenge: let rival teachers produce the origins of their churches and the succession of their bishops (Prescription Against Heretics 32).
The force of the argument is limited but real. A public chain does not automatically make every officeholder faithful, and the Fathers themselves opposed unworthy bishops. It does show that early Christians expected apostolic teaching to remain publicly traceable in visible churches rather than disappear into a secret restoration story.
Primary sources
1 Clement 42–44: https://www.newadvent.org/fathers/1010.htm
Irenaeus, Against Heresies 3.3: https://www.newadvent.org/fathers/0103303.htm
Tertullian, Prescription Against Heretics 32: https://www.newadvent.org/fathers/0311.htm