Since I originally posted this thread, I removed the original speedo and found the spindle was bent and not spinning freely around the drum (refer to @Mattax 's photo above). The spindle was contacting the housing causing the clicking and erratic needle movement. I attempted to re-shape it and it got better, but was still not right.
I replaced it in the car with a spare speedo I had on the shelf and unfortunately had a similar, but less severe issue, which is probably more like what @JAndrea posted above.
Trying to repair this mechanism myself was more than I wanted to take on. Props to you if you can fix it yourself, but it sounds like worn parts will need replacing and they do not all show the exact same failure mode (different parts could be the issue in any given case). Some combination of magnet shaft / upper jewel / speedcup / bushings were mentioned when I contacted several repair services.
The consensus from those communications was that these mechanisms were not a robust design and frequently wear out. I found a few places that sounded like they were experienced with restoring them. I ultimately sent both units to a place called "Mr Heaterbox" in Canada because they custom manufacture new bushings for these and offer full restoration, including the dial face and needles if you want those rehabbed. The prices were comparable to other places, but they were all in the range of $150-$250 per unit for service and calibration. Shipping to Canada was a pain and I am still waiting for the results, but I hope to have two restored units back in my hands soon. Thanks to everyone who replied with all of the valuable information and pictures.
Sounds like the OP has found a good speedometers repair shop, I will add one more.
I have had two speedometer heads repaired and cleaned by Deluxe Speedometer in Denver CO. I found their turn around time and pricing to be very acceptable. I am not at all affiliated with that business, passing this on as they did good work for me.
