My recollection for the demise of the name "Jehovah's Christian Witnesses" in JW literature is tied to when dozens of recorded message telephone lines by anti-JW groups started up in the mid-70s.
The first one, I believe, was in Phoenix, Arizona about 1975. They held up large signs outside a JW assembly advertising the phone number with the recorded message and got hundreds of phone calls. The next year the number got published in the phone book under "Jehovah's Witnesses" resulting in a large number of phone calls. However, the local JW authorities were not amused. They complained to the phone company and they were going to refuse to allow the number to be published in subsequent phone books. There was a lot of wrangling over this between the phone company and the Phoenician JWs and this anti-JW ministry. (The anti-JW ministry claimed they were Jehovah's Witnesses but not affiliated with the Watchtower Society.) The final resolution was to list the anti-JW phone line under a new heading: Jehovah's Christian Witnesses. The advantage of this was that alphabetically this meant that it appeared in the phone book before "Jehovah's Witnesses."
In those pre-internet days dozens of other recorded message lines were set up across the US and most also copied this nomenclature. So, the name "Jehovah's Christian Witnesses" came to be associated with these anti-JW groups and I think that's the main reason the WT Society decided to stop using it so that it wouldn't end up advertising these groups.