All perfect praise be to Allaah, the Lord of the worlds. I testify that there is none worthy of worship except Allaah and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.
In principle, a pilgrim is required to throw the pebbles by himself. It is invalid to delegate another to do this on one's behalf unless one has a valid excuse, such as sickness, and so on. In Al-Mughni, Ibn Qudaamah said, "If a man is sick, prevented by some reason, or has another excuse, then it is permissible for him to delegate another to throw the pebbles on his behalf."
Thus, if that old man is unable to throw the pebbles by himself due to aging or sickness, then it is permissible for him to delegate someone to do this ritual on his behalf. Slaughtering an animal is due on him in case he is able to do by himself but he delegated someone to do it on his behalf.
As for your case, it seems that staying with that old man is not a sound excuse to allow you to delegate someone to throw the pebbles on your behalf. It was possible for you to go to throw the pebbles after the return of your companions, or one of them, so that they could stay with the old man. So the ruling concerning a person who abandons throwing while he is able to do it is applicable to you. You are required to slaughter a sheep within the sacred precincts of Makkah and distribute its meat among the poor there. You can delegate someone to do this on your behalf.
Allaah Knows best.