Yes, it's not just homeless people with this bootstrapping problem. When I first arrived to the US in the nineties as a student I needed a social security number, for this I needed a P.O. Box (they did not accept the dorm house as address). For the P.O. Box I needed a social security number. Most international students ended up breaking the deadlock by making up a social security number.
I had a similar issue living abroad. My wife had a work visa (which was the reason we we moving) and I was allowed to go being a spouse, but once there getting a permit to work for myself was impossible without a job, and a job was impossible without a work permit.
There were ways around it, but it took finding a job at a really big company to make it work - they had dealt with it and had HR people that specialized in it. Once "on paper", I was pretty free to move around. I would not be surprised if their method was just putting in all zeros in the system or something until the permit number came back.
