Pretending to be Wise
To add to the other points made, the example of Israel v. the Gazans seems to be cherry-picked to me, as there are plenty of conflicts where the Great Powers have taken sides- i.e. basically any UN intervention, since they require the US's support. Even so, the Great Power pretty clearly has chosen aside in the Gazan conflict, since the US Senate passed a resolution officially endorsing Israel's side.
More generally, I think people, at least part of the time, do realize that being neutral is effectively the same as endorsing the stronger side, and simply remain neutral in order to deliberately pick sides- that is, they side with the stronger side, without having to pay any real cost for their decision. To give a real life counterexample to your examples, it's congressfolk voting 'present' instead of yes or no, which is effectively the same as voting no, but without the cost of signaling your disapproval, and so convincing people to dislike you.