If a person davens for himself and mentions his own merits it appears as if he is being haughty and demanding reward for his actions. This alone takes away from his merits as true prayer requires humility and for this reason Chizkiyah's prayer was found wanting. The correct method of prayer would be as we pray, requesting that the merit of our forefathers protect us (in the first Berachah of the Amidah) and only then making our own requests in the other Berachos.

There is a second issue here. A person who is in a predicament is deemed a "Chavush" -- a prisoner who cannot free himself and therefore requires the intervention of others (see Berachos 5b). Thus, a sick person requires other people's prayers and not just his own. Similarly, a person praying is in need and therefore requires the merit of others.

Yoel Domb