Maybe a bad ignition switch. From the FSM wiring diagram, the small purple taps off the big purple and connects to the GP relay. Jumpering them shouldn't do anything since they should be connected.

Purple/white appears to get power in START, but not run. If you get +12 there with no key in the ignition, that might be the problem. But since the starter isn't turning all the time and only when you jumpered it by removing the NSS, I'd also say the NSS is bad.

Bk/Red is probably back up lights power (upstream of the NSS), but FSM says Bk/Ylw/Red and there should be a Gray/Yellow (downstream of the NSS to the lights). Check for +12 @ Bk/Rd in start/run, check fuse 14. If no power in start/run that's another indicator of a bad ignition switch.

Summary from the FSM diagram (with the NSS connector connected):

Vi/Wh = +12 v in start, 0 other times
Fat Vi = continuity to Vi/Wh in P or N
Bk/Ylw/Red = +12 v in run/start (check fuse 14)
Gy/Ylw = continuity to Bk/Ylw/Red in R
Skinny Vi = connected to Fat Vi at the connector