Greetings!
I'm having trouble diagnosing a starting issue.With the cold weather and no garage and an intermittent issue running tests has been hard.

1983 240d automatic
starter may be original, can't tell?

The car starts normally most of the time. When it doesn't, I always hear the solenoid click, there doesn't seem to be a weather/temp/ usage pattern to when it fails.

It always starts if I short together the large (+) bolt on the solenoid to the small (+) lead on the solenoid. If I short the 2 large bolts it just sparks.

Sometimes it starts if I wiggle the gearshift, however, since I hear the solenoid clicking whenever it's in P or N and not the other gears i think the NSS is working OK. I think I'm just reengaging the solenoid when I do this. Backup lights are fine.

I tried bypassing the NSS by jumping the headlamp switch (+) terminal to the purple white pin under dash (to starter sol.) still just a click. I think that rules out the Ignition switch too? Wiggling the Ig Sw doesn't help.

Sometimes rolling the car a few feet then trying to start it again gets it to start. but that didn't help today.

Had battery / charging system / starter circuit test from Advance Auto (for what it's worth) all checked fine. Of course it was starting that day anyways. Battery is 8 months old.

I've cleaned (with battery terminal cleaner) and tightened chassis to engine ground strap. Batt (-) and chassis terminals and Batt (+) terminal.

With Voltmeter 12v from battery, 10v at solenoid (+) terminal when key in start position. This with analogue meter though so not sure of accuracy.

Having hard time tightening alternator belt (worn out tensioner)but it's not terrible.

So I would just put in a new starter and hope for the best but I'm concerned that since it always starts from the screwdriver trick as stated above.... that It might be something else

Any suggestions for a next step, anything I've missed or need to re-check?

Thanks!
Sean