[...] why not administer the vaccine earlier on (pre-approval) to the most vulnerable? if P(death|vaccine) < P(death|covid) then it seems really barbaric not to give it.

The full equation is: P(death+injury|vaccine) < P(death|covid)

Before widespread testing, you don't know the severity of a treatment's side effects, especially in vulnerable subpopulations like children and elderly. The risk of severe complications for a novel drug are especially of concern when the immune system is involved, since it's often unstable or senescent in early and late age groups, and overreaction (e.g. cytokine storms) may prove toxic.

You don't want the cure to be worse than the disease unless the untreated mortality rate is sky high and no other therapy is viable, as it is for many cancers. In the case of Covid, with a 2-10% death rate (depending on age and region), such desperate measures are hard to justify, especially in low risk groups, until risk of injury is better known.