KM & push button mechanical stop timers
I think John got it almost exactly spot on - 9 LK models had this type of timer from what I can figure. These are the 63, the 64, 65, 66, 68, the 70, 71, 72, and the 1974 Lady K. There wasn't a new model for 1967, 1969, 1973 or 1975. There were only tiny differences between the 70 and 71s, and the '74 was a large capacity re-do of the 1972 with one less speed. The 66 LK had a number of derivatives (like six or seven) which were in production as 900 and 800 series models into the early seventies, including Kevin's coppertone 800 machine which was a 1970 or 1971 model.
There were also some earlier models, including the elusive 1962 800, which though not an LK, was essentially featured just like the '63 but with 1962 console styling. There was also a 1961 Kenmore 70 with push-button timer stops, which I think is the lowest this went downline.
There were also some 24-inch machine variations too that had push-button timer stops. That line was always for niche markets and not nearly as commonly made, thus I am not sure any of us here have seen all the variants that KM had, but there were at least a couple. They usually used warmed-over styling from 29-inch KMs from one or two years before.
Interesting question Malcolm!
Gordon