Attempting to repair my hazard switch, my blinkers stopped working (it was while I was trying to test the contacts). Fuses are fine, so my guess is it's the relay. I've spent some time studying the S57, and think I have it correctly diagnosed, but wanted to confirm, as that particular diagram is quite involved.

The Green/Pink wire that enters the relay has 12v+ power when the car is on. However, when I plug it into the relay, the power goes away (grounding? maybe due to a stuck relay?), and does not transfer to the Green/Blue to power the blinker stalk switch.
Is that how it's supposed to work? Am I correct in thinking replacing the relay is the first step?

As always, help and insight is appreciated!

Yes it sounds as though there is a fault with the flasher unit.
Note: This is not a relay
It is a flasher unit. Basically a bi-metallic strip that makes and breaks the connection to the indicator switch as it heats and cools, based on the current flow through the circuit.
They are commonly available from auto parts shops - make sure you get the type that has a single terminal for the warning lamps.

Turns out it wasn’t the flasher unit…new one didn’t change anything, so I sprayed (more) contact cleaner and poked around in the hazard switch. Something must have been stuck or something because it works now. Magyvered a working hazard switch based on another forum member’s idea to use washers to hold it on. Not perfect, but it works for now!