Let's talk about why a VNM utility is useful in the first place. The first reason is prescriptive: if you don't have a VNM utility function, you risk being mugged by wandering Bayesians (similar to Dutch Book arguments). The second is descriptive: humans definitely aren't perfect VNM-rational agents, but it's very often a useful approximation. These two use-cases give different answers regarding the role of completeness.

First use-case: avoiding losing one's shirt to an unfriendly Bayesian, who I'll call Dr Malicious. The risk here is that, if we don't even have well-ordered preferences in some region of world-space, then Dr Malicious could push us into that region and then money-pump us. But this really only matters to the extent that someone might actually attempt to pull a Dr Malicious on us, and could feasibly push us into a region where we don't have well-ordered preferences. No one can feasibly push us into a world of peach ice-cream, and if they could, they'd probably have easier ways to make money than money-pumping us.

Second use-case: prediction based on approximate-VNM. Just like the first use-case, completeness really only matters over regions of world-space likely to come up in the problem at hand. If someone has no implicit utility outside that region, it usually won't matter for our predictions.

So to close: this is an instance of spherical cow in a vacuum. In general, the spherical-cow-vacuum assumption is useful right up until it isn't. Use common sense, remember that the real world does not perfectly follow the math, but the math is still really useful. You can add in corrections if and when you need them.

The VNM theorem is best understood as an operator that applies to a function $Comp(W_i,W_j)\to \{W_i<W_j,W_i>W_j,W_i\sim W_j\}$ that obeys the axioms and rewrites that function in the form $Comp(W_i,W_j)=Compare\left(E\right[U\left(W_i\right)],E[U\left(W_j\right)\left]\right)$ where U is the resulting "utility function" producing a real number. So it rewrites your function into one that compares "expected utilities".

To apply this to something in the real world, a human or an AI, one must decide exactly what $Comp$ refers to and how $(>,<,\sim )$ are interpreted.

• We can interpret $Comp$ as the actual revealed choices of the agent. Ie. when put in a position to take action to cause either $W_i$ or $W_j$ to happen, what do they do? If the agent's thinking doesn't terminate (within the allotted time), or it chooses randomly, we can interpret that as $\sim$. The possibilities are fully enumerated, so completeness holds. However, you will find that any real agent fails to obey some of the other axioms.
• We can interpret $Comp$ as the expressed preferences of the agent. That is to say, present the hypothetical and ask what the agent prefers. Then we say that $W_i<W_j$ if the agent says they prefer $W_j$; we say that $W_i>W_j$ if the agent says they prefer $W_i$, and we say that $W_i\sim W_j$ if the agent says they are equal or can't decide (within the allotted time). Again completeness holds, but you will again always find that some of the other axioms will fail.
• In the case of humans, we can interpret $Comp$ as some extrapolated volition of a particular human. In which case we say that $W_i<W_j$ if the person would choose $W_j$ if only they thought faster, knew more, were smarter, were more the person they wished they would be, etc. One might fancifully describe this as defining $Comp$ as the person's "true preferences". This is not a practical interpretation, since we don't know how to compute extrapolated volition in the general case. But it's perfectly mathematically valid, and it's not hard to see how it could be defined so that completeness holds. It's plausible that the other axioms could hold too -- most people consider the rationality axioms generally desirable to conform to, so "more the person they wished they would be" plausibly points in a direction that results in such rationality.
• For some AIs whose source code we have access to, we might be able to just read the source code and define $Comp$ using the actual code that computes preferences.

There are a lot of variables here. One could interpret the domain of $Comp$ as being a restricted set of lotteries. This is the likely interpretation in something like a psychology experiment where we are constrained to only asking about different flavours of ice cream or something. In that case the resulting utility function will only be valid in this particular restricted domain.

Yes, you’re missing something, but it’s not what you think.

Actual humans do not conform to the VNM axioms. You correctly point out one serious problem with axiom 1 (completeness); there are also serious problems with axiom 3 (continuity) and axiom 4 (independence). (It also happens to be an empirical fact that many humans violate axiom 2 (transitivity), but there’s much less reason to believe that such violations are anything but irrational, by any reasonable standard of rationality.)

Oskar Morgenstern himself regarded the VNM theorem as an interesting mathematical result with little practical application in the real world, much less application to actual humans. This is also my own view, and that of others I’ve spoken to. It is, in any case, not at all obvious that conformance to the axioms is mandatory for any rational agent (as is sometimes claimed).

The assumption which I advise you to discard is that the VNM theorem is either descriptive of real (or realistic) agents, or that it’s prescriptive for “rational” agents. It is neither; rather, it’s a precise mathematical result, and no more. Anything more we make of it requires importing additional assumptions, which the theorem certainly does not dictate.