Utility functions are not primitive. They are constructed using an algorithm specified by vN&M (or Savage, or A&A). Constructed from preferences over lotteries over outcomes. Preferences are primitive. Priors over states of nature are primitive. Utility functions are constructs. They are not arbitrary.

As has been mentioned, if you constrain preferences using one of the standard vN&M axioms, and if you assume that you can construct a lottery leading to any outcome, then you can prove that outcome utilities are bounded.

I think that the OP needs to be seen as a proposal for constraining the freedom to construct arbitrary lottery-probes. And, if the constraint is properly defined, we can have an algorithm that generates unbounded utilities, but not poorly behaved utilities - utilities which cannot be used to construct expectations that are not unconditionally convergent.