It seems to me that the state can mandate vaccination if they determine that the advantages outweigh the disadvantages, full well knowing that there may be (probably will be, in fact) some rare side effects.

How is this different than the state drafting people into the military, knowing that some of them will die?

In both cases, the state is obviously responsible for the outcomes. But that doesn't mean that there is a liability on the behalf of the state, or that you can hold an individual functionary working for the state responsible.