"Suppose the house price drops" is a very non-specific hypothetical. Why would the house price drop, and what impact could that have on homelessness?
A drop in housing prices doesn't immediately fix any of those things, but it unlocks some options. Looking at Zillow, rent of a 1 bedroom unit is around $2000/mo (put another way, that's 100% of the income of a person working at minimum wage), that's just an impossible situation.
If housing prices come down, rents come down, the most functional 1/3 of the homeless population gets off the street, resources can be applied to the more difficult cases, etc.
Not an economist. But many of us perceive that in the US today the middle class is contracting. A few people from that class may migrate to the upper class-- good for them-- but probably the overwhelming majority of folks migrate downwards.
So if OP means that in a context where there is a lot of quite expensive real estate you would expect also to find a good number of people who are so marginalized that they are living in their cars or on the streets, then yes, they are linked.
For those already homeless, special assistance will be required to get them back into "moneyed" society. However, a lower entry point to homeownership would make this assistance less costly.
What we can do is judge whether a policy helps or hurts. Higher housing costs causes more homelessness. Period.
If we're dealing in hypotheticals, why do you seem to be asserting that they would not do so? If they can hold down some sort of job, and they can afford the lower rent why wouldn't they go for a real roof over their head?
Or are you saying that campers simply don't want that? I have no idea what you're on about with these questions...
The other category of “invisible” homeless - those living in cars and at friends places/garages will of course benefit if the average price of the housing drops