You're missing the fact that how much Joe values the surgery depends on whether or not he expects to be told whether it worked afterward.

Good point.

If on the other hand his utility-of-belief function were U(p)=sqrt(p), then being told the information would be worth ((1/3)sqrt(1) + (2/3)sqrt(0)) - sqrt(1/3) = -0.244, so he plugs his ears.

I agree here.

But I still suspect that if your U(p) is anything other than linear on p, you can get Dutch-booked. I'll try to come back with a proof, or at least an argument.