At the time of the Buddha people left the household life for the wandering life of a mendicant, some staying beneath trees or in caves, but even at those times, nuns and monks also had accommodation, like individual huts or shared dormitories where they stayed. During the Rains retreat monks and nuns are required to stay in the one place and have a hut with a lockable door. It’s true there may have been a hard core culture of wanderers who lived and slept rough, but there was definitely accommodation provided, as the suttas and Vinaya clearly show.
Perhaps instead of thinking about this concept literally, you could consider it as leaving the home, leaving the household life. Not having any household duties or responsibilities, not living that life anymore. Not acting or behaving like a householder in terms of storing up food or possessions, not cooking or things like that.
These days, we should probably only use the word ‘homeless’ to refer to those who are living in the streets, in cars, couch surfing etc. as we understand it today is nothing like what it meant in the Buddha’s time. It is a problem of translation. Eg:
Since I’ve gone forth
Yadā ahaṃ pabbajita
from the lay life to homelessness,
I think the most important part of the phrase “gone forth from home to homelessness” that we often use in English, is the “gone forth” part, as it indicates the transition to a new, ordained status.
Meanwhile, real homelessness is a really big problem and as you point out, getting worse. While It’s great that people provide accommodation for monastics, it should be simple and not luxurious.
Further, although the Sangha is mostly well supported, there is still a big disparity between the amount of accommodation for female monastics that we need to address.
Generally, it’s wonderful if people give to homeless shelters, lobby governments to do more and help work towards dismantling the systems that lead to homelessness, which is a cause of suffering for many.