The Ampex 1" type C video tape recorder needed is rare, but it's not impossible to find one. The NSA could certainly watch the video if they wanted to. The just don't want to go through the effort for a FOIA request.
Yes, they do show up on ebay, but usually not in working condition. Then you have to find someone that can do a restoration. Keep in mind that there may only be one chance to play the tape before it falls apart, so the player needs to be working perfectly.
Digitizing is the easy part. Even though it's an ancient format, it's still just NTSC composite video and analog audio. You digitize it just like you would a VHS tape.
The zero-effort method of pointing a camera and mic at the screen as it's being played back should be sufficient if they can't do it another way. Given the tape's age, the resolution is unlikely to be high enough to lose significant details that way.