So this issue is solved. It was a two part problem.
About 3 months ago (on top of the banging issue), the refrigerator started only intermittently cooling. It was still banging and clicking as it was before. I suspect the cooling issue was masked when the weather was cold and the kitchen was relatively cool. But once the weather became warm, this generally coincided with an obvious inability to cool. I didn't track the temperatures before though. I only recall they were acceptable, but... can't say if they were in as tight a range as they are now.
Edit: If you have this problem, it would be useful to get a refrigerator and freezer thermometers. If the freezer temp not tracking tightly around 0F, and refrigerator temp is not tightly within 32F and 40F, it could indicate a temperature sensing issue.
The repairmen came about two months ago and replaced the "start device and capacitor." This quieted down the banging by about 85%. Instead of banging loudly every day, and commonly having loud clicks from time to time, it would happen once a week, and it was muted. And it was more of a chugging I guess. The loud click would occur with the banging/chugging too. The refrigerator still wasn't cooling consistently though. It would intermittently cool, getting down to freezing, but it would routinely stray well above freezing in the freezer and up to nearly 70 in the refrigerator.
Then about a month later (I don't have much perishable food, was able to get by, and I waited to gather more evidence and try to identify any other patterns), they replaced the "refrigerator thermostat". THIS totally solved the problem, in conjunction with the starter mechanism fix. It's been nearly a month, I haven't heard any clicks, no banging/chugging at all, and it's holding 0F in the freezer and 32-40F in the refrigerator very well, and the compressor starts and runs smoothly.
So, I think the root cause of the click/banging was a malformed start signal maybe? That signal going into a worn starting mechanism causing the banging, and then that click which coincided with the compressor banging stopping.
FYI. @jerry-contrary and @ed-beal, thanks for the information.