Pyruvate as a final electron acceptor in the electron transport chain
Someone else may come along later with a definitive answer but I found this question intriguing so here are my thoughts:

The standard redox potentials of the mitochondrial ETC carriers are:

NAD⁺/NADH              -0.32 V
complex I (Fe-S)       -0.27 V
complex II (cyt b₅₆₀)  -0.08 V
complex III ((cyt c₁)  +0.23 V
complex IV (cyt a₃)    +0.38 V
O₂/H₂O                 +0.82 V


Note that electron transport is carried out from negative to positive.

Now, for pyruvate/lactate the standard redox potential is -0.19 V so on the basis of this pyruvate could in theory accept electrons from complex I, but not any further down the chain. However, complex I normally transfers electrons to coenzyme Q within the membrane. The standard redox potential of CoQ is +0.04 V which is already too positive to be able to then reduce pyruvate at the membrane surface. Thus there would have to be a novel way of transferring electrons from the final FeS centre of complex I, which is within the membrane, to pyruvate. Pyruvate is, of course, soluble and is generated by cytoplasmic glycolysis, so this transfer would have be at the membrane surface that faces the intermembrane space of the mitochondria.

If such a scheme had evolved then it might be possible to achieve some proton pumping through the novel version of complex I, which would be  energetically advantageous. 

Another complication is – assuming that a complex I with pyruvate as electron acceptor had evolved – how would the cell/ mitochondrion regulate electron flow during aerobic respiration? Presumably the pyruvate-accepting version of complex I would have to be kept inactive somehow until needed.

Clearly it is much simpler to have a soluble lactate dehydrogenase to deal with the pyruvate and regenerate NAD⁺.