How does cell surface oxygen consumption happen? Here is the main reaction overall (it's probably not that simple!):
4e- + 4H+ + O2 -> 2H2O
A certain amount of hydrogen peroxide is also generated and just a little superoxide (probably for cell-cell signalling). Other electron acceptors can stand in for, or compete with, oxygen as the terminal electron acceptor. The reaction occurs on the outer surface of the plasma membrane.
The source of the electrons is cytoplasmic NADH, which is oxidised to NAD+ and H+ on the inner surface of the plasma membrane. Each of the electrons is transported via a plasma membrane ubiquinone:semi ubiquinone cycle which then donates them to the redox enzyme on the external cell surface. Here is a simplified version of Herst and Berridge's Figure 1:
Although I doubt that there is very much NADH or NAD+ in normal extra cellular fluid there has to be an NADH docking site on the outer redox enzyme as one of the hallmarks of cell surface oxygen consumption is that it can be halted completely by flooding the cell culture medium with NADH. Electrons from this then follow the dashed line to fully reduce the ubiquinone to ubiquinol and the system grinds to a halt.
So you can measure cell surface oxygen usage by the fall in consumption which occurs when you add exogenous NADH just as you can measure mitochondrial oxygen usage by the fall in consumption which occurs when you add myxothiazol.
Why is the system there?
Let's go back to the two routes of glycolysis. Without the glycerophosphate shuttle (insulin signalling driven/driving) we have
Glucose -> lactate -> mitochondria -> pyruvate -> TCA
and there is no depletion of cytoplasmic NAD+ as one is consumed and one produced in the glucose -> lactate process. With insulin signalling we have two parallel processes:
Glucose -> glycerophosphate shuttle -> CoQ -> ETC
which consumes cytoplasmic NADH, leaving none to convert pyruvate to lactate. So in parallel we have to abort glycolysis at pyruvate:
Glucose -> pyruvate -> mitochondria -> TCA
which balances the cytoplasmic NADH:NAD+ ratio nicely.
Now, let's consider a cell undergoing rapid growth with a view to divide. For today I will ignore mitochondrial biosynthesis and consider what happens if cytoplasmic pyruvate is consumed for amino acid biosynthesis. For each molecule of pyruvate which has been diverted to an amino acid there will be one less available to provide cytoplasmic NAD+ by conversion to lactate, which will limit glycolysis because cytoplasmic NAD+ is essential for the oxidation of glyceraldehyde-3-phosphate.
Under these conditions cell surface oxygen consumption appears to be able to step in to oxidise cytoplasmic NADH to cytoplasmic NAD+, which then allows glycolysis and its associated ATP production. This looks to be particularly important if there is any sort of a problem with the ETC and the glycerophosphate shuttle.
In rho zero cells, where the ETC is deleted (and there is no glycerophosphate shuttle) and glycolysis is the sole source of ATP production, cell surface oxygen consumption has to supply NAD+ in direct proportion to how much pyruvate is lost to anabolism rather than being used to supply NAD+ via lactate generation. In rho zero anabolic cancer cells cell surface oxygen consumption can be as much as 90% of the total oxygen consumption of the parent cell line.
TLDR: Anabolism requires cell surface oxygen consumption to regenerate NAD+. Its importance rises if there is defective ox-phos.