I had to move out of an apartment a few years ago and into a spare half-story at a friend's house. The apartment was attempting to raise my rent, while not to 100% of my income (yet), a significant amount. Their excuse was "that's the market rate now," and when I told them to F off and went to check other nearby apartments, they were correct, that was the "market rate."

I have absolutely zero doubt that if the government had paid me the difference they were trying to charge (as UBI or any other stipend), they would've raised the rate again by the same amount very quickly. In fact, going by the idea that supply & demand are what caused that (rather than simple landlord greed that I generally see it as), UBI definitely can't fix it, since UBI will not increase supply or decrease demand-- they'll still need to charge just enough that some people can't afford it in order for there to be "enough."

You're somewhat correct in that the solution (building more housing, I guess) is not really related to UBI.