Affordable and available housing has everything to do with homelessness though, it's one of the best ways to actually keep people from becoming homeless in the first place. If more people can afford a place to live, less people will be homeless. Won't fix all of it but a huge chunk anyhow

I have no idea if or how much old Eastern Bloc countries lied about the number of homeless. I wouldn't be surprised at all, but I haven't seen any studies or statistics about this so I can't assume they were all lying or that the situation was universally worse than in Western countries.