Why is there continuity between hot and neutral in the unmodified part of my circuit?

Long story. I’ll try to make it short:

Goal: Replace cooktop exhaust fan/light

I’ve done this before. Should be no big deal… Removed old fan/light. Started to install new fan/light. Connected cable – armored (BX?). To make sure I have the grounding right, I checked continuity from metal case of fan/light to armor and to neutral. All good. But then checked to hot and that also showed continuity, not good.

I figured the problem was the old (~65 years old) cable with insulation falling off the wires. Went to a big orange store to get something to cut the armor, new bushings, etc. Cut off some of the armor (fortunately this cable had some length to spare) to get some good wire – same problem! Figured it must be down the line. Next fixture is overhead light. Removed fixture to get to the wires. Disconnected the cable going to the fan/light (that’s all good) and problem is coming from cable from the switch box.

Opened up the switch box and found a big mess. Two switches, one cable coming in, one cable (3 wires black hot/red switched hot/white neutral) to overhead light and fan/light, one cable for second switch light fixture and one cable going on to at least two rooms of the house. Hot wire looped around a screw and then one to another screw instead of using a pigtail. At least one wire around a screw the wrong direction. Etc. Cleaned it up, marking wires as I went with colored tape (I’m learning) since all the old wires are shades of gray.

Isolated the problem to the ongoing cable. Capped the two wires of that cable.

Put overhead light and fan/light back together – all worked. Having a bit of trouble getting the switch box (but with the bad cable wires capped) back together, but I’m sure I can get that done.

The question: Since everything was working when I turned off power to this circuit to start this "little" project, and I found the problem (hot shorted to neutral in a cable far from where I was working), any idea this would happen? And any idea how to find the source of the problem, or is it just the usual "keep taking things apart until you find the problem"?

Resolution: Ed Beal and Harper’s comments were 100% correct. Unplugged everything and the phantom short went away. The interesting thing though is that some of the devices (the kitchen LED light fixture and the new kitchen fan/LED light fixture that started this journey) did not cause a phantom short. In fact, leaving those lights on let me have an easy way to check (yell upstairs after restoring power "did the kitchen lights go on?"). So this is not an absolute "must turn off/unplug everything". But oh what a mess…turns out downstairs refrigerator & freezer on the same circuit and I hadn’t realized it. Freezer definitely OK, refrigerator almost certainly lost some stuff. Ugh. Live ‘n learn.