TL;DR
I found that the simmer burner was severely caked up with something that overflowed at some point so I gave it a boiling water bath with Dawn dish soap.
I think the spark wasn't making proper contact with the metal burner and causing an arc which was shorting the clock. The weird part is that any knob activates all sparks at once so this symptom should have been present regardless of burner. ¯\(ツ)/¯
With the help of this answer, I finally got off my butt and opened up the stove top.
The burner is working properly now so I think I'm all set but below are all the things I did:
- Turn off gas and unplug stove
- Remove grates
- Remove burner caps (soak in boiling water)
- Remove burners (soak in boiling water)
- Unscrew gas nozzles from stove top
- Remove front knobs (soak in boiling water)
- Remove front knob cover plate using 4 screws from below
- Remove 2 screws below the edge of the stove top
- This stove does not use push-tabs as advertised in some videos I watched
- Push gas nozzles down under the stove top
- Slide stove top towards self and lift up
- Remove and re-insert all spark wire leads from distribution box
- Clean gas orifices using a needle
- Mine didn't seem dirty but I'm already in this deep =)
- Do not use a toothpick because the wood can break off and cause a gas clog
- Scrub down all the boiled items with a stiff bristled dish brush and dry them off
- Re-install everything