Let's imagine there was somehow zero rental market. Imagine there was a law against purchasing a dwelling and then not actually using it as your residence. People still need to live somewhere, so there would be a demand for housing. People would see a profit in meeting that demand, so someone would build and sell housing. Currently, those who can't afford to buy a home have rental as a cheap alternative. Without that, there would be an open niche for something to meet the need for housing. There would be a market pressure to solve the discrepancy between the price of housing and the available capital of the average person. House prices might be forced down, salaries might be forced up, I don't know what would happen precisely but there would be a pressure to make it possible for people to live somewhere.

You can see evidence for this in what happened in a lot of major cities. People have been able to use one home that they own as collateral in buying a second, and then use the income from renting it out to pay that off plus a little profit. That leaves them with two properties as collateral and a little cash spare, making it easier to do it again with a more expensive place. Rinse and repeat and you've got wealthy landlords buying up all the properties so there's no need for the people selling those properties to drop prices to where first-time buyers can afford them - the usual dynamics of supply and demand that keep prices in reach of buyers have been disrupted, and the two types of buyer separate into two tiers that get pushed further apart, getting harder and harder for people to jump from the lower tier to the upper. This is how you end up with people paying £1000 in rent while the bank tells them they can't have a £700-a-month mortgage because they can't afford it, and that £1000 a month leaves them nothing left over to save up for the £30,000 deposit they'd need anyway. The market pressure that led to this situation are obvious, and reversing those pressures is the most obvious way to fix the situation.

I can't speak for OP but I think in general the idea that "landlords shouldn't exist" or whatever doesn't just stop at eliminating rentals. There is more than enough housing in the US to house everyone here (and probably the world? if not, there's enough resources to build housing for everyone), and it seems unjust to let people be homeless or exploit their need for shelter due to artificial scarcity.

I like to think that most of us could agree that everyone deserves the dignity of having shelter regardless of what luck life has dealt them, even if we can't immediately agree on how.

The abscence of landlords does not preclude the existence of housing. The house would still be there if there wasn't a landlord attached to it.