Itt seems to me that it is actually easy to define a function $u’(...)>=0$ such that the preferences are represented by $E(u’^2)$ and not by $E(u’)$: just take u’=sqrt(u), and you can do the same for any value of the exponent, so the expectation does not play a special role in the theorem, you can replace it with any $L^p$ norm.
Itt seems to me that it is actually easy to define a function $u’(...)>=0$ such that the preferences are represented by $E(u’^2)$ and not by $E(u’)$: just take u’=sqrt(u), and you can do the same for any value of the exponent, so the expectation does not play a special role in the theorem, you can replace it with any $L^p$ norm.