Something I want to note about homeless shelters-in many districts in the US homeless people have no rights to nondiscrimination in shelters and discrimination is rampant.
Some courts will insist that homeless shelters don’t count as housing or as public accommodations and don’t apply the non-discrimination laws that would affect even something like an ordinary restaurant (though those are grossly under-enforced in general). There are shelters who have received public money that have been legally allowed to discriminate in regards to things like gender, race, sexuality, trans status, religion, and disability.
Even if homeless people had the sort of resources to sue for equal access in court, which they don’t in reality have in any substantial way at all, there are large swaths of the US where they would just be told the law doesn’t apply to them.
Homeless shelters are a major area of discrimination in the US. It’s not uncommon for LGBT people, POC (especially Black people), disabled people, non-Christians, drug users, women, mothers who are homeless with their children, teenage boys/teenage male assigned at birth kids trying to be housed with their mothers and siblings (this usually happens to children of color far more than white children), domestic abuse victims, immigrants, leftists, people who speak English as a second language or have limited English, and many others to face pretty intense discrimination in accessing even the extremely limited amount of homeless shelters that do exist. And the law absolutely fails them in a myriad of ways over and over again.
If I as a queer trans person tried to access a shelter, including ones that take public funds, and they just flat out told me they don’t serve LGBT people it’s not even clearly illegal in the US, just as it wouldn’t be clearly illegal for them to refuse to make the building one I could physically use as a disabled person. While I haven’t lived in a shelter, I’ve heard numerous horror stories from people I knew personally, and the closest shelter to where I live (which is still over an hour drive away, which, how would I even get there?) is one that refuses to serve people like me.
While there are some very good shelter workers out there and some programs that really are respectful and caring to the people they are supposed to serve, abuse, paternalistic control, theocratic bullshit, and discrimination are extremely common. And the social and legal systems typically choose to support and enable that abuse. This system is no substitute for a real robust public emergency housing program, let alone a full housing or adequate public housing system.