Not true.
There is no one way it “would actually play out”. The person you were talking to was acting like there was only one valid way to think about housing (and about the imaginary idea of rental housing not existing for some unexplained reason), and they knew both what it was, and what would happen, and that’s a bullshit argument.
For example, one thing that actually is happening in some US communities, is that housing prices rise toward what SOME people are willing to pay for housing… and when/where there are enough people with particularly high incomes, rents rise to the point that there are very few available rentals that are affordable by people with low incomes, and that causes and/or perpetuates homelessness. And in those cases, people who care, look for ways to create affordable housing, sometimes to be offered to homeless people as part of a program.
Now, you haven’t really said whether you would count charitable low-income housing managed by the government or other not-so-landlord-like organizations to be what you’re talking about, or not.
If you mean “zero landlords of any kind”, then you’re going to have to look to a very different sort of situation, and may have a hard time finding one in a modern economy.
My own utopian solution would involve gradual rent control measures, then phasing out rental arrangements, and also phasing out property taxes, in favor of the idea that everyone should have a right to live someplace, and that for-maximum-profit rental and real estate deals are antithetical, because they lead to ever-rising prices and pervasive scarcity and homelessness. But to support that would require an economic revolution of one sort or another.