In your post you said “we can’t trust the government with the keys to backdoors”. I think that we can probably go even further than that, and say that we really cannot trust any government with safely or sanely controlling any part of any solution to the problem of governmental access to encrypted devices.
This is an incredibly important and useful realization, because it now allows us to disregard any proposed solution in which the government has control over its use. Some other entity needs to be in charge. This is particularly important when you consider that any such solution will have to work globally. Solving the problem in the context of a well-governed and well-behaved ‘western’ democracy is one thing, solving it in dictatorships and countries where the rule of law is nothing but a carefully constructed illusion is another thing altogether.
However, creating an entity that is able to operate wholly independently of the laws of its staff’s countries would be a huge challenge. However, a virtual organization, with a globally distributed staff, connected by hidden services hosted on the ‘dark web’ might be a possible solution.