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endoself comments on Unconditionally Convergent Expected Utility - Less Wrong Discussion
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The fact that this is a problem does not make anything in the post novel. In the grandparent, I linked to discussions of this problem that touched on everything that you discussed here.
Since utility functions are only unique modulo affine transforms, you can't combine them using naive expected utility. The correct method to do so is unknown.
I'm aware of this, but fail to see how it would change the ability to make probability distributions over the space of utility functions and then take expectations there. Sure, you'd be doing it over equivalence classes of functions, but that's hardly any difficulty. What I am saying is you can assign utility to choices of utility functions: utility functions must inherently be recursive in practice. And so their non-summability (or other technical difficulties) causes immediate problems.
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.
You had one link for changing the expected utility just to make Pascal's mugging go away, and another that seems to be based on the same idea, but has flawed reasoning and a different conclusion.
The first link was to the comment, not the post; I disagree with the post. The proposal in the second link was qualitatively similar to yours and it failed for the same reason.