This would lead eventually to the system we have now, only larger (because it's not means tested). It also has huge supply-side economics problems:
If you stipulate that every family can spend an extra $1000 on housing, rent will go up by $1000 pretty quickly, and housing prices will go up by whatever $1000/mo in mortgage payments will allow.
In consumption-based positional goods, rents will always go up by whatever the basic income is. Because people are entering a zero-sum rat race for status. That's irrelevant to whether it's doled out as BI or as housing voucher.
But in normal markets, people will still try to supply the good at the lowest cost and consumers will still try to gain it at the lowest price, and the market will find some midpoint.
Moreover, if you can't make "universal housing vouchers" work, how are you going to make "universal basic income" work? The former is a much simpler problem.
If you stipulate that every family can spend an extra $1000 on housing, rent will go up by $1000 pretty quickly, and housing prices will go up by whatever $1000/mo in mortgage payments will allow.