The correct solution should be something in the middle. Old-fashioned wiretapping, with a warrant and the need to dedicate staff to installing and monitoring the tap is basically okay. The problem is that the mathematics of cryptography and the scaling inherent to information technology mean that only all-or-nothing solutions are possible. If the cryptography is intentionally broken, it's broken not just for law enforcement, it's broken for everyone. If law enforcement has a backdoor they can use with a warrant, they're capable of using it without a warrant, and probably will. And if their special keys get leaked, then again, the encryption is broken for everyone.
Like you point out, secret sharing is one way of getting around this in principle. But governments would never make their access dependent on an NGO; in practice I'm sure they'd only agree to secret sharing schemes where the separate parties were separated only by nominal bureaucratic firewalls, and then you're back to the original problem.
