To what principle would I properly attribute the concept that the current in the hot and neutral conductors must balance?
It would be the "principle" of the GFCI's particular circuit design (probably more than one way to execute a GFCI).
Whether the node is a solder joint on a pc board or a skyscraper in downtown Chicago the principle holds.
Since you mention Kirchhoff there are several variants on his law, one for voltage around a loop must add up to zero, which is obvious because the voltage across a node must be zero, and another for all current into a node must equal the current leaving a node... Also obvious since the node can not create or destroy current.
There are probably multiple examples in nature similar to Kirchoff's current law node, (like the water flowing into a water pipe tee junction will equal the water flowing out of that tee... the tee can't create or destroy water) but an important given for his current law is we are talking about a single node (his voltage law applies to voltages around a loop with multiple nodes but that is not remotely like a GFCI either).
I find that a lot of people are confused about electricity because they don't understand the basic physics involved-principles that to someone with your background are as taken for granted as the law of gravity-and that are always true. I did not quote the law-nor do I feel "name dropping" out of place as anyone reading this can easily find more in depth information if they so desire.
I am probably being pedantic, but I object to the mis-application of the well known law about currents in a single circuit node, to a more complex circuit better characterized as two conductors in series with the hot and return from an AC power source, to sense if any current that goes out to the load, does not come back. Any such lost current is considered a potential shock hazard.
My advice for lurkers is to not get distracted by all my pedantry.. All we need to know about GFCI is that they compare the current flowing out of the hot lead into the load, to the current returning from the load in the neutral lead. If any current is lost from the load and does not find it's way home, the GFCI trips.
JR