Nearly every country on earth strugle with homelessness. In the UK they have seen a 4 % increase yearly increase in homelessness.
People sleeping rough and being homeless is a huge problem. There is also a large majority of people that are homeless, but not registerede as one because they are sofa surfing or has been offered something they just didn’t feel safe in.
And if they manage to get homelessness to 0% who is going to pay for it? Finding jobs and homes to such a large number of people will cost millions of pounds.
And the only way to finance that is through the taxes which means they either have to save some money on other parts of budget or increase the taxes.
Increasing taxes will mean they will take a larger cut on the citizents income, and those on minimum wage that is barely getting by now will have even less money. And possible send some minimum wage workers on the streets.