Directly, o2 provides the final electron in the process. Indirectly, it enables an entry for water because of its chemical properties.
Could you not breathe? Yes. Oxygen doesn’t directly enter any step of mitochondrial electron transport; it just mimics a happening that would happen for bacteria elsewhere were there insufficient oxygen (i.e., when we use nitrate in water to make nitrite). Think of it as a place-holder that enables other reactions to take place, and so indirectly facilitates mitochondrial electron transport.