I'm not sure what you mean formally by these assumptions, but I don't think we're making all of them. Certainly we aren't assuming things are normally distributed - the post is in large part about how things change when we *stop* assuming normality! I also don't think we're making any assumptions with respect to additivity; is more of a notational or definitional choice, though as we've noted in the post it's a framing that one could think doesn't carve reality at the joints. (Perhaps you meant something different by additivity, though - feel...

27mo

I wasn't saying you made all those assumption, I was trying to imagine an empirical scenario to get your assumptions, and the first thing to come to my mind produced even stricter ones.
I do realize now that I messed up my comment when I wrote
Here there should not be Normality, just additivity and independence, in the sense of U−V⊥V. Sorry.
I do agree you could probably obtain similar-looking results with relaxed versions of the assumptions.
However, the same way U−V⊥V seems quite specific to me, and you would need to make a convincing case that this is what you get in some realistic cases to make your theorem look useful, I expect this will continue to apply for whatever relaxed condition you can find that allows you to make a theorem.
Example: if you said "I made a version of the theorem assuming there exists f such that f(U,V)⊥V for f in some class of functions", I'd still ask "and in what realistic situations does such a setup arise, and why?"

An example of the sort of strengthening I wouldn't be surprised to see is something like "If V is not too badly behaved in the following ways, and for all v∈R we have [some light-tailedness condition] on the conditional distribution (X|V=v), then catastrophic Goodhart doesn't happen." This seems relaxed enough that you could actually encounter it in practice.