Well, turning off BTMM doesn't help managing remote Airport basestations, especially when Manage over WAN is not turned on (as per good security practice). Using BTMM to a remote OS X system works to access a remote Airport basestation if, and only if, there happens to be a remote OS X system to connect to and run Airport Utility.
This problem occurs with Airport Utility 6.3.6 and 6.3.7 on 10.11.6 and 10.12.x and with Airport firmware 7.6.4, 7.7.3, and 7.7.7. Additional symptoms include disappearance of printers shared using an Airport Extreme.
However, screen sharing access to remote OS X systems remains viable. I have tested using Airport Utility on systems accessed through Screen Sharing and found identical symptoms.
My conjecture, based on geographically diverse systems and software versions is that there has been some change in apples wide area bonjour service which is not successfully communicating with the Airport basestations. This would mean that Apple is the direct source of the problem, not any user boxes.