But yeah, #2 is what I'm afraid of too. Paraphrasing a cartoon villain, if everyone is wealthy, nobody is.
Besides, in the past decade, cost of living / housing / rent has gone up so much that even a $1000 / month basic income can't give you anywhere decent to live anywhere. In addition to basic income, we need basic housing - which is dangerous, because it invokes the USSR's rows of depressing and substandard apartment buildings. But everybody should be able to live comfortably at a standard of living. Everybody should be able to have access to and afford a two bedroom house or apartment on a single income, or the social safety net if they are not employed.
2) It's not that simple because different goods will respond differently. Certain goods will get cheaper because the increased sales will allow for more economies of scale.
Also, most of the homeless are unhoused because they either can't afford a home or they have mental illness/drug addiction issues that makes them incapable of earning money in order to be able to afford a house. It doesn't matter if housing is vacant if you don't have the means to pay the rent, or if they won't even consider you for a lease anyway because you don't have a reliable source of income.
My argument is that we don't know what the overall effect will be.
Edit: In other words, if housing increases by $10 but the cost of other stuff decreases by $20 then you still come out ahead.
The income tax code already exists and has to solve some of these problems, so it seems easier to give everyone the money and then tax it back from the highest earners later (or implement it as a negative income tax, but that has its own hurdles as well, i.e. imagine a homeless person needing to wait until tax season and then getting paid for the whole year, it would be a big hurdle and they'd still need other assistance programs the rest of the year if they didn't budget the money well enough).
Do you also expect food, transportation, entertainment, and technology costs to go up $1000/month nationwide?
Rent goes up due to a lack of available apartments or houses in the area. If public policy is geared toward allowing development of sufficient housing for the people wanting to live in a place, that will have a far bigger impact on rents than UBI.
Rent is the one you are sort of locked into. Food, etc, you have choices ... moving, much harder to shop around.
I worry that if you give everyone $1000/month:
1) you cannot live on $1000/month so it is an empty gesture
2) rent goes up $1000/month nationwide so it is an even emptier gesture