Does energy come from fuel or oxygen? (surprise?)
[OP:] It is usually said that fuel contains energy and that oxygen only enables the release of energy in the sense like enzymes enable reactions.

An enzyme is a catalyst, so it does not change the enthalpy of a reaction. That part is correct. &quot;Oxygen only enables the release of energy&quot; is incorrect. Oxygen is one of the reactants, and the oxygen atoms also appear as products.

[Dr. Schmidt-Rohr in JChemEd paper] Generally, our analysis indicates that it is not the organic fuel but rather O2 that is “energy-rich”.

This might be true in the context the combustion reaction with the set of organic molecules he chose to include. The organic fuels exclusively had C-H, C-C, C=C, O-H, C-O and C=O bonds, if I understand correctly from the abstract and the figure. He is saying that the bond energy of most bonds in reactants and products are similar (e.g. C-H vs O-H, C=C vs C=O, 2 C-C vs C=O), and that the O=O &quot;double bond&quot; is an outlier.
I think it is worthwhile to state the amazing fact that we have a biradical (see https://chemistry.stackexchange.com/a/15061) in our atmosphere and are breathing in a very reactive substance. The only reason that our red blood cells aren't constantly undergoing mutations from oxidative damage is that they lack DNA once they are released into the bloodstream.
If I were to study the combustion reactions with elemental metals, I might come to a different conclusion than the paper. Try comparing $\ce{Na + O2}$ with $\ce{Hg + O2}$.
Final remark (quoting myself):

When a raw egg rolls off a countertop and falls on the floor, does it crash because the countertop is so high or because the floor is so low?