I suppose the simple explanation is that it required the commanding officer, science officer, and chief engineer. Note that in LTBYLB Spock does not identify himself as First Officer, but as Science Officer. This is echoed in TSFS where Chekov identifies himself to the computer as Acting Science Officer.
So what if one of these department heads is missing/dead? Interestingly, in the novel Vuclan's Heart, during the battle of Narendra III, Garrett notes at one grim point that she cannot activate the self destruct of the Enterprise C with her science officer and chief engineer dead.