Why is it important to remove the used up electrons in oxidative phosphorylation?
I get that oxygen is the terminal electron acceptor in oxidative phosphorylation. But why is it essential to accept these electrons and remove them as H2O?


A redox reaction is made of two half reactions. Neither one of them is more or less important; the combination of both determines the energetics of the reaction.


  Why is having an efficient garbage collector (oxygen picking up the used electrons) so crucial to cellular respiration, and to ATP production?


The electrons are not "used" as in an old car or worn-out clothes. There as shiny and new as before. Without oxygen, the reaction would not happen in the first place (electrons are transferred from less tightly bound to more tightly bound). The Gibbs energy of proton transport matches what is necessary for ATP synthesis, which in turn matches the Gibbs energy of the redox reaction. With a different electron acceptor (specificities of the enzymes involved aside), the proton gradient might be too weak to make ATP.