While that'd be able to guarantee housing on an "a home exists for every person in the country" level, it pretty often means sending people away from economically-healthy areas (which should have plenty of housing competition / are or will be built to capacity, because they're healthy). Subsidized housing in those areas is more expensive by far for someone, regardless of how it's done.

Hence the wait-lists even where it exists; people want to live where they can be near what they need (be that work, medical, family, etc), not literally anywhere. The "literally anywhere" tradeoff may be worse than homelessness, or at least perceived to be.

Does not seem to be true based on all the waiting lists and shortages of government funded housing in the US.

Assume a well funded program. Everyone gets a home at an affordable rate, and homes are built in anticipation of need.

Why would you assume that? I would assume a program funded by Congress, and therefore a political football.