Obama: cryptographers who don't believe in magic ponies are "fetishists," "absolutists"

I share your skepticism about whether current government leaders would implement such a system well, but as a speculative exercise I think it’s worth considering–if we had a set of leaders who wanted to ensure the possibility of searching computers of suspects in certain specific crimes like terrorism and child pornography, but who took the concerns about the key being hacked or leaked extremely seriously, is it even possible to imagine a technically feasible solution they could implement that would satisfy both competing demands? Is there any reason the key would have to be “available for law enforcement agencies across the nation”, or why it would have to be on an “open internet with billions of them”?

My first stab at coming up with such a solution would go like this. Let’s assume the number of requests is small enough you could just have one or a small number of facilities that are allowed to use the secret key, and only if the actual physical device that needs to be broken into is sent there, after having had its ability to connect to the internet physically disabled. Then whatever high-level official is responsible for breaking into the device has to make sure the device is disconnected from all internet connections, and they enter the secret key which is printed on a piece of paper (or something more durable like a piece of metal) kept inside some ordinarily locked device akin to the “nuclear football”. The relevant data on the device can be transferred to another (offline) computer, and then the device itself must be destroyed just in case it contained some secret software or hardware that could record the keystrokes when the secret key was being entered (for further security we could imagine the whole process, from entering the key into the device up to the device’s destruction, is done inside a Faraday cage just in case the device had some kind of secret transmitter the authorities missed). Is there anything obviously impossible or unsecure about this on a technical level, again leaving out the likelihood of present-day government officials being motivated to come up with a solution this careful?