A lot of homeless people do end up in prison both for all of the anti-homeless laws in various cities. For some folks it may end up being a better option than just outright dying in the streets from lack of food, medicine (love how the richest nation can’t put our tax money into anything that actually helps all of us), and/or from the elements. So while your point of bringing up prison population is a valid one. It is still (in part) related to homeless population numbers, just not totally useful by itself.
That being said, I do think that China is actually putting much more effort into dealing with the problem. As bringing people out of poverty/homelessness is crucial to keeping support of the masses. When enough people are just tossed out and can’t get help. It will lead to dramatic and very widespread revolts and even civil wars. That is true for any nation, no matter what system is in place. Socialist nations were birthed by such widespread poverty for the many while the few hoarded the wealth and resources. It is kind of one of the main points Marx and other anti-capitalist theorists base their shit on.