A Phoenix isn't immortal because it can withstand everything that's thrown against it. That's a great strategy for a mountain, but not for a living thing (even a mythological one, apparently). It's not indestructible. It endures what it can and then burns to ashes. It dies - in one of the most agonizing ways imaginable. The Phoenix's ability is that it suffers its painful, fiery destruction and is then reborn.
And the new Phoenix? From what I've found, its rise from the ashes isn't as majestic as it's often pictured (see userpic!). It's reborn small, weak, and frail - just strong enough to gather the ashes of its previous incarnation, put them into an egg, and carry the egg to Heliopolis. It may be a beautiful process to the outside observer, but from the Phoenix's perspective I suspect it's a horrific experience (though better than the alternative).
I've gotta say, I know the feeling. Also, I may not be able to fly on my own, but I've received an incredible outpouring of love & support from people willing to help carry me when I need it. For that I am deeply, deeply thankful.