I've done some additional testing and now think I may have been facing at least two problems simultaneously.
First, I replaced the fuel filter. The old one looked like it'd been around a while. To start the mower for additional testing (described a bit below), I used the choke for the initial startup and only once after it died out. I then ran smoothly for ~20 minutes before I turned it off.
Second, in tracking down the purple wire, I discovered that the run relay would click when I messed with the wires to the ignition switch, while the switch was in the off position. I pulled the harness off the switch and found the connector to be damaged, particularly connection S1:

This connector and S1 on the switch look like they suffered from some high current:

I found I could hold the connector in such a way that the mower would start right up (when connected up to the ignition switch). In fact, I got it to start and let it run for ~20 minutes with no problems. This suggests the connector has a short somewhere. I tested continuity on the ignition switch, and it worked properly for off, run, and start (p. 26 in
https://apache.ariens.com/manuals/04983500A.pdf).
When I tried to start the mower yesterday, it seemed sluggish. I tested the battery, which is only a few months old, and found it at 11.7 V. So I charged it in preparation for today's testing (described above). I wonder if the problem with ignition-switch connector may have lead to the battery running down?
Do you agree with me that the main problem I'm having appears to be related to the wiring harness connector to the ignition switch?
Replacing the entire harness is an expensive and time-consuming task. Any suggestions for a work-around? My current plan is to cut out the connector and directly connect the wires to the switch, carefully insulating each with heat-shrink tubing and electrical tape. I'd splice on additional wire if needed.
But this leaves me with the concern about what caused the over-current. Any ideas of where I should look for the source of the problem? Or other problems this source may have caused?
Thanks again!