Most people don't quite understand how homelessness works, and it's because of this whole "support network" thing. In most people's mind, homeless = bum living in a cardboard box. What actually happens in reality is someone falls on hard times and they move in with a relative, or bum a room from a friend, or etc, etc. That's what keeps them out of the cardboard box, and keeps them fed.
And that's why they're shit scared to move. Move, and all those friends and family can't help you. Even scarier, move, and you're probably scared the whole friends-and-family ties might get weaker - most people are always paranoid about whether their friends/lovers/etc still care about them, so they stay close to make sure they're "tending the fire" and keeping the friendship alive. It's human.
At the same time as supporting the cities they live in doing all of the most fundamental work (cooking, cleaning, etc) they are regularly disrespected and told to fuck off somewhere else if they're struggling - away from the support network they rely upon and the friends and family who give their life meaning.
And yeah, some of them did move away but that sometimes made things worse because even though they might be financially slightly better situated in a different city, they didn't have a support network there.
Expensive housing is a political choice. Watering down the minimum wage was a political choice. These people falling through the cracks and suffering was the outcome of those deliberate choices to prioritize pumping up asset prices and profit margins over actual people.