An unconfirmed nominee doing nothing to advance the administ

There is no ambiguity. Consider the following:

The ice cream will melt unless eaten quickly. In this case, it's clear that the ice cream is what needs to be eaten quickly.

The ice cream will turn to liquid unless eaten quickly. Now we have two nouns--ice cream and liquid. Is it unclear which needs to be eaten quickly? No. Even aside from a meaning perspective, we can't apply "unless" to "liquid" because it is the object of the action and so doesn't fit the structure of "X will Y unless Z." Note that we could also put the "unless" phrase up front (ignoring "because" in the original):

Unless eaten quickly, the ice cream will turn to liquid.
Unless confirmed, nominees do nothing to advance the administration's goals.

Try to create a correct sentence in which that "unless" phrase applies to the object or a noun in a modifying phrase, and you should find that you can't do it!

I won't let anyone in unless 21. No--unless they are 21.
You can't see out the window unless cleaned. No--unless it is cleaned.

Or consider this:
The sauce will not pour smoothly from the bottle unless shaken vigorously.

This might seem especially ambiguous--are we shaking the sauce or the bottle? Maybe it doesn't matter, since we're clearly going to do both at the same time. True enough, but structurally speaking, "unless shaken" applies to the sauce. To see this, let's build up the sentence bit by bit:

The sauce will not do Y unless Z.
The sauce will not do Y unless shaken thoroughly.
The sauce will not pour smoothly unless shaken thoroughly.
The sauce will not pour smoothly (from the bottle) unless shaken thoroughly.

Similarly, we can build up the structure of our original clause:

Nominees do Y unless Z.
Nominees do Y unless confirmed.
Nominees do nothing unless confirmed.
Nominees do nothing (to advance the administration's goals) unless confirmed.