Per Retrophile's suggested tip involving my relay being wired incorrectly, I changed the wiring and figured out my mistake (thanks Retrophile) - I assumed incorrectly that the relay's pins fit logically to the pin layout in the schematic... they don't. Going to the documentation for this particular relay verifies that Retrophile's pin layout is correct.
The two photos here show the rewired circuit - same circuit but just a slight change in photo angle for you to get a better look.
So, my next step was to shoot another short video showing me using the logic probe on the 555 chip. My description here may not be 100% on the money, but as I understand the chip and the schematic, pushing a switch causes pin 2 to detect a drop in voltage... and causing pin 3 to allow current to flow (or is it just an increase in voltage - is voltage always present? Something to look into...)
As you can see from the video, putting the logic probe on pin 3 and pushing the power button causes the probe to change in pitch... that should be right. But then when I press any switch, pin 3 should change in voltage and be detected by the probe... but nothing happens. Any thoughts?