Doug,
I've got an RCA CTC-16A with exactly the same grease film problem on a 21FBP22. Some day I'll get it cleaned. The whole chassis has to come out to remove the CRT so it is a pain in the ass job that is easy to put off.
I do not feel comfortable handling large 70 degree CRT's like these either. The imploding glass can do a lot of damage to the person carrying the tube if it goes thump. You really should wear eye protection or better a full face shield, a leather welder's apron or coat and gauntlet gloves IMO.
One tube that really frightens me is the 27 inch B&W in my 1953 Admiral.
BTW, did you happen to take a picture of the naked front of the Zenith CRT without the safety plate that you could post? You said it was green?