If the magnet inside the solenoid is intact, it should work. Report back to HQ. BTW, the adjustment on the throw of the plunger is preset. no need to touch it... .
Ok, for anyone combing the archives for the solution, I think I have it.
That adjuster you mentioned, while it may adjust the throw, also adjusts the timing of the engagement of current-to-the-starter-switch to the engagement of the pinion. As the contacts in the solenoid wear (or erode) away from use, the internal mechanism that closes the circuit to the starter motor can cease making contact.
In my case, a slight adjustment of the fulcrum allowed the internal contacts work every time.
I'm going to take it to the local starter/alternator repair shop to confirm.