There were some important qualifications to STIII's "The needs of the one outweigh the needs of the many."
First, besides McCoy, everyone else who went with Kirk on the stolen Enterprise
was a volunteer
. Kirk expressed "The needs of the one..." to Spock, not to postulate a sweeping counter-axiom to Spock's "The needs of the many...", but rather to let Spock know how much he in particular meant to them specifically. It was as if Kirk had said, "But we love you, Spock."
Second, neither Kirk nor Spock demanded that anyone die. Kirk intended that everyone live, in fact. Additionally, it was fortuitous for both Saavik and Spock that they even came at all.