I don’t think “progressives” have any issue with housing the homeless. The issue is where.
Go to a conservative (or indeed any) neighbourhood and tell them you’ll be building 200 apartments nearby to house rough sleepers, see how that goes down.
Most homeless are invisible to us anyway. They hold down jobs, they have gym memberships, they just sleep in their car, or on a mate’s sofa every so often. Nobody would have a problem with them moving in nearby.
It’s the aggressive beggars, addicts, and shitting in shop doorways (and these three are the same person) that nobody wants anywhere near them. These are who most people think of when they hear the word “homeless”. Most of them need more treatment than just a roof. We don’t have enough of that either.
I’ve no issue with my taxes helping all these people. I’m happy to pay tax to reduce the chances of me personally being robbed by somebody in desperate poverty.