I see the point the author is trying to make, but in my view (disclaimer: not an economist) the presentation lacks rigor and can be misleading. If you adjust oil prices for the economic output that you get out of a barrel of oil, then you no longer have a price per barrel. You have the price of the oil needed to produce X dollars of GDP, which can be a useful metric, but you should no longer call it a price per barrel as what you have effectively done is replacing barrels with a different unit of oil (that varies in time depending on oil intensity) in the calculation.

The result that making that adjustment stabilizes the prices (discounting the peak in the 80s) is not surprising to me. As oil prices rise, the economy adapts to use less oil, so the adjusted metric tends to an equilibrium. Isn't that what supply and demand is supposed to do?