And it's a problem, not just an "experience".
Sure, homelessness is a problem.
So is hepatitis, so is asthma, so is epilepsy, so is a noisy neighbour. Do I need to highlight that epilepsy is a problem when seeing someone experiencing it, or should we all already be aware that it is a problem?
And only certain people have that problem.
Sure, and only certain people have hepatitis, asthma, epilepsy, noisy neighbours etc.
They are not as wonderfully "equal" as you pretend, regardless of what rights they may hypothetically have.
??? Seriously? You think they are, what, sub-human? Not deserving of all the rights and privileges that society otherwise affords
all people? Which rights would you take away from them?
They have a special problem which needs special treatment.
So do people with hepatitis, asthma, epilepsy, a noisy neighbour etc.
By emphasizing that they're "just like everybody else", you're disrespecting their problem.
I would instead say that you are highlighting that it
is a problem, but not disrespecting them by equating their problem with who they are: they are a person (respecting) with/experiencing a problem (highlighting), they are not "the problem".
Maybe there are better phrases than "person
experiencing homelessness", such as "person with no home", or some such, but the point is that the phrase used should not describe the person
as their problem, any more than we should describe people by any other problem they may have. E.g. better to say "people with blindness" rather than call them "the blind". The former stresses that they are a person with a particular disability, rather than labelling them
as their disability.