I think you have it backwards. It's not about max_a U(E(a)), but of max_a E[U(a)]. Otherwise insurance wouldn't work. Let's illustrate with an example. Suppose

I think you have it backwards. It's not about max_a U(E(a)), but of max_a E[U(a)]. Otherwise insurance wouldn't work. Let's illustrate with an example. Suppose you buy theft-insurance, there's 10% chance of being robbed and the cost of robbery is $100000. Then, E[a] = 0.1 * -100000 = -10,000. So you'd be maximizing U(-10000). This is different from maximizing E[U(a)] because in this case it's 0.1 * U(-100000) + 0.9 * U(0). It's different if you are not completely neutral.