It seems that even if I have some algorithm that is on the surface not maximizing expected utility, such as being risk-averse in some way dealing with money, then I'm really just maximizing the expected value of a non-obvious utility function.

Not all decision algorithms are utility-maximising algorithms. If this were not so, the axioms of the VNM theorem would not be necessary. But they are necessary: the conclusion requires the axioms, and when axioms are dropped, decision algorithms violating the conclusion exist.

For example, suppose that given a choice between A and B it chooses A; between B and C it chooses B; between C and A it chooses C. No utility function describes this decision algorithm. Suppose that given a choice between A and B it never makes a choice. No utility function describes this decision algorithm.

Another way that a decision algorithm can fail to have an associated utility function is by lying outside the ontology of the VNM theorem. The VNM theorem treats only of decisions over probability distributions of outcomes. Decisions can be made over many other things. And what is an "outcome"? Can it be anything less than the complete state of the agent's entire positive light-cone? If not, it is practically impossible to calculate with; but if it can be smaller, what counts as an outcome and what does not?

Here is another decision algorithm. It is the one implemented by a room thermostat. It has two possible actions: turn the heating on, or turn the heating off. It has two sensors: one for the actual temperature and one for the set-point temperature. Its decisions are given by this algorithm: if the temperature falls 0.5 degrees below the set point, turn the heating on; if it rises 0.5 degrees above the set-point, turn the heating off. Exercise: what relationship holds between this system, the VNM theorem, and utility functions?