This is the problem with housing subsidies: they very often go to those who don't need them. In New York, your choice of apartment is primarily based on location -- anything that's not in the >$1 million range is pretty similar. So if you're looking at two apartments near your work and the rent stabilized one is $1200/mo and the other is $3000/mo, you try to get the $1200/mo one even if you could afford the $3000/mo…
The downside, of course, is that the cost of such payments is far more transparent to the public. Many economists would argue that it's in fact much cheaper, as you it's more efficient. Still, today's subsidies are well hidden and almost never show up in household budgets. And thus it's hard to get to a system of direct payment.