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From a mechanical standpoint, I don't like the 87V switch wear but it was making good contact and still better than most of the meters I have looked at.
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As an 87V owner I was also surprised and concerned by the awful rough movement developing in the rotary switch, although it's difficult to make sense of what is happening (how it 'feels') based on the sound as reproduced on the video. I gave it some thought and suspect it is the result of two things: the spring contacts being slightly 'loose' in their housing (by 'loose' I mean the tension holding it in place may be lower than optimal), and secondly the fact you use a constant speed when turning the switch on your jig. I would suggest the latter, in combination with the less than secure spring retention, allowed a sympathetic chattering to develop.
Chattering will be dependant on a number of factors: the amount of friction between spring contact face with plated track, the surface area of the contact face (which will increase as small flats are worn into them as they bed in), and the base spring tension pressing the two components together (oh, and speed!). If the spring retention mechanism allows enough flexibility such that friction overcomes its ability to slide smoothly (and thus sticks for a fraction of a second) then you will get sympathetic chattering, however this will be exacerbated by using a constant speed allowing the pattern to be imprinted into the track itself. Once established there will be positive feedback and things will only get worse as the procedure continues. Note that a lubricated track may have reduced the friction to the point where sticking would not occur - but all the other factors need to be in place before this would become a critical factor.
(There are similarities with using a parting tool in the lathe, the factors that cause chattering are similar, and if nothing can be done about the stiffness of tool mounting or play in headstock bearings then adding lubrication and/or adjusting cutting speed often resolves the problem)
Such perfectly constant speed provided by your test jig is not something that is seen in real life when simply turning the switch manually, so the chances of developing such a pattern of chattering will therefore be reduced. I do understand your idea about reducing the possibility of heat being generated due to turning the switch too rapidly but I also think you should vary the speed too.
It may be too late for that test example now (given the state of the tracks), but perhaps re-tensioning the spring contacts so they are held in place more firmly, a few seconds on a buffing wheel to round off any sharp flats formed on the spring contact faces, and perhaps a little lubrication would restore proper function.