I tried to distinguish between SSH the multiplexed secure network channel provider and Secure Shell, the default and overwhelmingly popular use case of SSH. Theyre different, but generally conflated. I have no real problem with the transport mechanism.
To answer your questions you probably have a dedicated on host agent. See MCollective or AWS SSM for examples Im familiar with. It doesnt really matter that the transport or message model is. Its more important that your actions are discrete, well modeled, with defined & structured inputs and outputs. Its also inportant that theres an intermediary so you can get the scope of control away relying on client and end hosts.
For #1 interactive is almost orthoganal. For #1 & #2 Its that you have structure, repeatability, validation, auditing, testability, etc. #3 again a transport problem, or similar.
For your last point my issue is if you treat dev and prod as similar change and access problems. The environment can certainly look or feel the same. But the output of dev should be discrete managed artifacts. If you need to do adhoc exploratory stuff on prod thats a good sign you lack control or valid synoptic model of your systems.
That said, we may be trying to solve different problems. I have lots of teams, more ICs, and many hosts. Getting from artifice to industry is important at some point.
Edit: go check out the mcollective demos to get an idea of how you can change the access/use model.