It wasn't directly from this answer, but this answer got me in the right direction. For the first time since the fall of 2017 (no time to troubleshoot this last year) I'll be able to drive my car again!
I'm looking at a wiring diagram for my 64 Galaxie. Hopefully your Comet is similar. It looks like the brake and turn signal filiments are the same ones, so if the turn signals are working then I agree that eliminates problems at the back end of the car.
I would start by unplugging both wires to the brake switch on the master cylinder. A cheap test light with a wire clip and a pointed probe would be the ideal tool here but volt meter can be used as well. Clip the lead to the engine, maybe a carb bolt. Touch the probe to the + terminal of the battery. It should light of course. Now use the probe to test each of the two wires that went to the brake switch. One of them should turn on the test light (the green w/red stripe in my case). If the light won't go on, figure out why.
Maybe a bad fuse, or loose fuse clips.
If the light does go on, then try connecting the two brake switch wires together using a test lead or small bolt or something. This bypasses the brake switch. If the brake lights are now on, it is likely a bad switch. If not, leave the two wires jumpered together and move inside the car. Use the test light to see if the green wire (in my case) going into the turn signal switch has power to it (from the brake light switch).
I checked with the above method and my test light turned on, and then I was able to successfully bypass the brake switch to see the lights light up. (Aside: I had previously tried to bypass the brake switch to no avail leading me to believe that maybe I actually had two problems.)
This had me scratching my head because my multimeter said that the switch had continuity. Of course the switch is in a horrible place for removal, so I hadn't yet pulled it off. That's when it hit me. The wiring was fine, the switch was fine - I bet the switch isn't getting activated. Last year, my brake fluid had gotten very very low due to a leak. I had refilled it and the brakes were fine, but I thought well, maybe there is some air in the line. BINGO! I removed it, had my son push the brake to get some fluid to push through, then I reinstalled the switch while he had the pedal down. Viola. Pressure & brake lights. This doesn't explain to me why the brake pedal had any influence at all on any previous outcomes, but hey, I can live with the mystery if I can drive my car
Thanks to everyone who chimed in.
For future generations searching for this answer, I hope this was helpful for you. Did I need to replace my turn signal switch? Maybe, maybe not, but it kinda needed it anyway, so I'm not disappointed I did that.
Next up, fixing that slow brake fluid leak that was the root of the problem, but that's for a different weekend. Tonight, we drive.