We see here Telemachus‘ astonished reply to Athena/Mentes. Suddenly he wants to get his guest somewhere they can genuinely speak privately so that he can ask lots of questions. Telemachus also tells us that his mother Penelope has never stopped believing that his father Odysseus is alive.
Immediately we see Telemachus thinking of his duty to his father. Yes, one might wonder why Telemachus has not done more until now to assert his authority, especially if he thought his father dead– in which case by right he would be king. My take is that Telemachus had grown up more or less with the suitors and accepted at some level their claims. In Greek society, a would-be royal heir who was too young or inexperienced to lead and rule did not necessarily inherit. So Telemachus might very well doubt his own right to kingship. After all, even if Odysseus was certainly dead (which would always be in doubt) what had Telemachus done yet to gain κλέος and thereby the clout to claim the kingship?
More than anything else, getting Telemachus out and searching for his father Odysseus will establish him as a worthy leader. He will travel among kings and be accepted as an equal.
- Penelope (variety.com)