While objectively correct, the policy you describe is incredibly cruel. People often end up homeless through no fault of their own, and more importantly,
temporarily homeless. Should someone lose all their pets just because they end up homeless for a couple days or a week? Do we want to fill the animal shelters up even further and send more animals off to be euthanized?
Imagine being a young child, already traumatized by ending up homeless, and now the pet you've been growing up with gets dragged off by the authorities to be executed.
It's absolutely the case that difficult problems can require the state to exert force to solve them, but I'm also just not convinced that forcing people into shelters is going to fix everything. Especially in this case where we know the area does not have enough beds for everyone, and the constraints enforced to get a bed are onerous (throw out your belongings, get rid of your pets, etc)
Some of the people who reject housing also cannot realistically be in a shelter because they will just cause trouble for everyone else if you force them to go. My brother used to work in law enforcement and this was an issue with the most troublesome homeless people in the area - as a law enforcement officer he really didn't have any workable tools. If you take them to a shelter they'll get kicked out because they're troubled, and if you take them to jail their condition will not improve (only deteriorate) and then they end up back on the street again once they're discharged.