I think you are agreeing with the parent comment here even if you dont think so. These are the people that would take new inexpensive housing - people living with roommates or parents that want their own place, people commuting from far away. These people aren't homeless.
You _assume_ homeless are not among that population, but I would argue there are. Obviously not 100%, but we are arguing about moving the supply curve up, so there is a corresponding adjustment as we move along the demand curve.
But the core point I’m making is that the difference between enough rental units to move an average rental price of $2500/mo to $1000/mo means there are A TON of new units and it would reduce rent pressure even for rental units that are cheaper than $1000/mo.
The fact is that “mental illness” is super common, has many aspects (some are significant hurdles to employment and others are less significant. “Unemployable” changes over time as employers look at the cost trade offs and the hiring market.
And Section 8 vouchers (one of the largest programs to increase housing stability) already pay $x00/mo, so there is absolutely some dollar about above $100/mo where additional rental supply does start to reduce the homeless population.
Those were the people I’m speaking about, those are people that many of us see daily, and those are a large portion of the homeless wrecking havoc.
People already work low paying jobs in SF. They are likely living with multiple roommates or commuting from far away. Given a large influx of cheap housing would absolutely put downward pressure on housing insecurity.
Also, your perception of homeless is probably limited to the visible population that is chronically homeless perhaps in tents. Most homelessness is acute and occurs when rental prices increase, temporary unemployment, during bankruptcy, or when the renter/owner kicks out a roommate. These homeless tend to live out of their car or try to couch surf often. Acute homeless tend to have much better odds of finding housing again (maybe not in SF and maybe not during the worst of economic downturns). Lowering rent costs by 60% absolutely would help reduce acute homelessness.