For a rational agent with goals that don't include "being averse to risk", risk aversion is a bias. The correct decision theory acts on expected utility, with utility of outcomes and probability of outcomes factored apart and calculated separately. Risk aversion does not factor them.

"Risk Aversion," as a technical term, means that the utility function is concave with respect to its input, like in thelittledoctor's example. I think you're thinking of something else, like the certainty effect. But I don't know of anyone who considers the certainty effect to be a terminal goal rather than an instrumental one (woo, I don't have to compute probabilities!).

A proper utilitarian would feel approximately the same desire to do something about each

And we should be proper utilitarians... why?

what if we discover, for example, that some murders are not calculated defections, but failures of self control caused by a bad upbringing and lack of education.

Then we have evidence they will strike again.

What if we then further discover that there is a two-month training course that has a high success rate of turning murderers into productive members of society.

Does that exist? My impression is that violent criminals often have suffered head injuries, not just poor upbringings.

Even if it does exist, and we have a way to restore people to normalcy, are there strong game theoretic reasons to? There could still be calculated defections, which we should attempt to deter by punishing violent crime. The rehab program also seems far more useful before crimes happen, rather than after.