Why is there a need for food stamps? Food is not expensive. Isn't a rent the largest expense? So wouldn't it make more sense to give a Rent Stamps instead? I am sure people need free rent, not a free food.
But like with the tax code, the government (with no small influence from the voters) prefers to keep some measure of control—no candy bars! extra benefits at farmers markets! Though as you might imagine that control is illusory given most spending is fungible.
Before you say "that's why the gov should build the housing", it's important to remember that centrally planned things are generally worse than market planned ones, it's a bitter lesson but history shows it time and time again. Instead of rentees choosing how to spend their money the gov, who only cares in an indirect way, will be making that decision for them. Inevitably someone in this compulsory taxation-gov-built-housing loop will realize their bargaining position and create insurmountable corruption. What would a rentee do, swap local govs to pay taxes to in their city?
If you want your gov to help your situation, use the gov to allow people to build more housing. If that's not an option, you gotta start new somewhere else.