Etymologically, why does 見合せ ("exchanging a look") mean "interruption, postponement"?
No source I consulted tries to explain why it can have that nuance. One of the other meanings is "to exchange troubled looks; to face each other due to shared fear or surprise" (Nihon Hōgen Daijiten; WordNet Ruigo). So one possible derivation could be: a situation that's surprising or threatening, so that it makes people exchange glances > (semantic narrowing) > a situation that requires postponement and consideration.
However, I don't think it's the most probable one. The Nihon Kokugo Daijiten explains miawase like this:
To suspend the execution [of something], and look at [=consider, evaluate] the situation for a while (しばらく様子を見ること) (first attested 1702).
Also for the verb form, mi-awaseru:
To suspend for a while the execution of an action that was about to be performed, and watch over/watch attentively the situation for a while (しばらく様子を見守る) (first attested 1688).
So the implication seems to be that the situation is the thing being watched. The agent isn't looking at other participants, he or she is looking at the matter that was postponed, and carefully considering it. But then why the auxiliary verb -awaseru?
Another sense of mi-awase can shed light on this question:
To examine this and that by comparing one to another. To examine in contrast (見比べる). To juxtapose for clarification (照合する).
Similarly for the verb form:
To compare the matter at hand with another thing so as to inspect (=look at) them [in contrast to one another].
So you see, it's not -awase in the sense of doing something reciprocally with another agent. In this case it's not actually "to exchange a look". It's -awase in the sense of putting objects together for some purpose; like tsuke-awase = garnish (e.g. steak with leaf greens), or kō-awase = incense-matching (game where the players compare incenses and try to identify each one), or kotae-awase = checking answers (for your homework, etc).
I believe that, given the Kokugo Daijiten glosses, the "compare and contrast" nuance is the likely source of the "postpone" nuance. That is, mi-awase = to compare things in order to weight matters carefully > (semantic shift) > to stop and take the time to analyse.