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.

Similarly, you need ID to get an ID. You also need proof of address to get an ID. And you need an ID to get an address or a job.