Whistleblowing and legal affairs come to mind -- whistleblowers need to feel secure in order to report misdeeds, and even elected officials have attorney privileges.
Those certainly aren't the only cases, but they're two examples where open probably isn't the right default for communications.
Governors, Politicians, etc. are public servants. They are not there for their own good, they are there to follow the law, and the serve the interests of their constituents, and only that. Therefore it stands to reason that any communications they have with other officials, about official business, should be recorded as it is a matter of public record. It stops "he said she said" disputes. It helps us know about underhanded deals, unfair advantages, and many of the other things that people do when given such powers with no oversight.
Even with perfect life recorders I think you'd end up with a moribund apparatus playing 5 dimensional tic tac toe (that is, sensible people would call for letting them turn the damn things off whenever they wanted to).
If anyone wants to help out in collecting government communications using FOIA, please let me know. It's surprisingly easy to get bulk communications data of cities, though it ends up being very time consuming with everything considered. I could definitely use the help, especially with a project beyond the city level that started just this week.
Here's [1] an example of a dataset of ~2mo of email meta - about six million emails - I recently received from Houston, TX. (Thanks to Jeff Reichman at januaryadvisors.com for cleaning the data up!)
[1] https://data.world/sketchcity/city-of-houston-email-metadata...
Like, they can use self-disappearing messages, but all messages are automatically archived and encrypted with a pre-set password. I guess this could still open the data to being stolen if the password is discovered, but I don't really see any other solution to making both self-disappearing messages and FOIA requests working at the same time.
I'm a Signal user and hugely appreciate their push to bring encryption to the masses.
But when it comes to the government, I believe the ideal situation is that there is a literal live-stream of everything they are doing, at all times. Government officials are public servants. Sure, if you're discussing an ongoing military operation, keep it behind closed doors temporarily. But the vast majority of what they're doing should be made accessible to the public in real-time.
The only reason not to do so is to hide lobbyism, partisanship, and corruption from the public eye.
Sometimes it's vital that people be able to communicate in private with their reps.
This might already be the case and is much better then making senate use non-e2e secure communication software.