Why do you have to insist that 1/3^^^3 human lives is an insultingly high price for green socks?

At no point have I insisted that, or anything analogous. But that's not the question at hand. I haven't found any of my values to be non-Archimedean. The normative question is whether they should be outlawed for rational agents. From the post:

I'd say you're allowed to choose the green socks over the red socks every time, while exhibiting no willingness to sacrifice Big context expected utility for it, and still be considered "rational".

What's your take? Do you think it should be normatively illegal to have non-Archimedian values (like say, a pair of Hausner utilities) and be considered rational? Please share you thoughts on this thread if you're interested.

I don't see why the son-daughter-car formulation I described isn't good enough as is

As far as I'm concerned, it is good enough. So then the challenge is to provide reasonable assumptions which

1. allow the kind of analysis you describe, and

2. still manage to imply mathematical theorems comparable in strength to those of VNM.

VNM utility theory isn't just an intuitive model. It's special: it's actually a consistent mathematical theory, with theorems. And the task of weakening assumptions while maintaining strong theorems is formidable. Luckily, Bolker has done this, by axiomatizing expectation itself, so when we want rigorous theorems to fall back on while reasoning as you describe, they're to be found.

And I don't see why "objective randomness" or anything else needs to come into the picture

AFAIC, it doesn't. But whether we like it or not, the VNM theory needs the source of randomness to not be a source of utility in order for the proof of the VNM utility theorem to work. I find this unsatisfactory, as you seem to. Luckily, Bolker's theory doesn't require this, which is great. Instead, Bolker pays a different price:

• his utility is no longer unique up to two constants, as it is VNM.
• probabilities themselves become behaviorally ambiguous.

This doesn't bother me much, nor you probably; I consider it a price easily worth paying to pass from VNM to Bolker.

I admit this comment is not as in-depth an explanation of some concepts as I'd like; if I find it ties together with enough interesting topics, and I think it will, then I'll write a top level post better explaining this stuff.