Breathing's Role in Aerobic Respiration
There's a fairly large misconception here. The huge amount of energy that's available from respiration, and which drives the synthesis of ATP, in fact comes from that reaction that you consider secondary, and that's the transfer of electrons to oxygen.

If you want to think of it as two separate reactions, then the first one
$$Glucose + 12\ {NAD}^+ + 6 H2O ⇌ 6 CO_2 + 12 NADH$$ releases about 10 times less energy than the second one: $$6 O_2 + 12 NADH ⇌ 12 NAD^+ + 12 H_2O$$

The reason you get something resembling 30 ATP from a glucose molecule is because oxygen is such a great electron acceptor.

Many microbes can in fact use molecules other than oxygen as the electron acceptor, but in general it is much less energetically efficient than using oxygen.

You could try equilibrator for estimating the free energy available from various chemical reactions. First reaction and second reaction from above as examples.