If you want to live overall, and would still want to live if you lose something, it's not worth dying to save it.

The actual conclusion from these premises is that things are not worth dying for if you'd still want to live after losing them. Which is valid, taken as is.

But, following from your premises, if I wouldn't want to live after losing something, I would sacrifice myself. For example, I would not want to live in eternal pain, and would thus kill myself.

The problem is that there is a difference between a friend dying from cancer, and you deciding not to sacrifice yourself to save your friend from, say, a guillotine. Because while in the former, you're sad and grieving, in the latter, you feel like you murdered him. For some, that guilt seems like an unbearable burden, and so they decide, in the moment, that they don't want to live that way. So they die, despite the fact that they would otherwise be able to live without that friend.
This can be generalized. For instance, I live in Toronto, Ontario. I know that if I do not sacrifice myself, all of San Francisco will perish. For me, this is different from SF simply perishing on its own, because the choice is now mine to make; mine appears to me to be the choice between murder, and suicide. Of the two, choosing murder leads to a lifetime of guilt over killing someone (despite the obvious fact that I did it for self-preservation). This guilt may/may not be too much, and it is this which determines the choice between murder and suicide.

There is also the type of situation where, for instance, you either kill yourself, or your entire municipality gets killed, including you. The choice here is between your death, and the combined deaths of you and other people. Logically, since you die anyway (and there are no other options), you woukd sacrifice yourself.