The problem is that welfare systems, such as those that provide housing, that distinguish who is eligible by how much they can afford it, to a certain degree, inevitably depress higher levels of economic activity, and good saving behavior, through the very nature by which they're operated.

If we say that someone is no longer eligible for free housing if they earn, say, $2,000+ a month, and housing would otherwise cost $500 a month, then if they're currently earning $1,500 a month (the same they'd effectively have if they had to pay $500 a month for housing on a $2,000 a month salary) they have a direct incentive to not make over $2,000, unless they can guarantee they'll make at least that much plus $500 more to compensate for the difference. If they earned $1,800 a month, they'd be making $300 more than someone making $2,000, but paying $500 a month for housing after hitting the cutoff.

This isn't just a hypothetical either. While this 2021 study does mention some benefits of means testing, such as more targeted expenditure, it ultimately shows that...

"An asset means-test incentivizes low-income households to hold few financial assets making them vulnerable to predictable and unpredictable income changes."

...and sees that, in the end, while it can marginally increase the cost of these social programs to the taxpayer, it ultimately does more to benefit the individuals receiving the assistance.

Or how about this research done by the Cleveland branch of the Federal Reserve that states:

"the elimination of testing limits, such as in policies similar to a UBI, could present a welfare-improving alternative to the current system, though not without large economic trade-offs." (They effectively mean worse targeting of funds, but better overall results)

Means-testing directly reduces the incentives that lead to higher overall household wealth, and quality of life.

Not having means-testing increases total income, which also means increased tax revenue. That same tax revenue can then go to funding the housing system as a whole, but it won't directly, substantively punish people for an increase in income past an arbitrary threshold.

Not to mention the increased administrative cost of performing means testing, as opposed to doing unconditional support, which could reduce the amount of money actually going to funding housing, in favor of funding jobs for people that audit income levels of housing applicants.