People keep making more people, so who builds their houses?

Developers. If there is more demand in a market, they will build property, then sell it to whoever is willing to buy, or, will seek funding from an existing institution, which if it’s not landlords, will be housing cooperatives, then use that funding to finance new buildings. Traditionally, when we’re talking high-density housing, the buyer of these properties is a landlord. Without that landlord, the demand still exists, and someone, or some group of people, will inevitably fund the cost of the housing. In a world with no for-profit landlords, housing cooperatives fill in the gaps. (primarily for high-density housing specifically)

Either existing cooperative members come to an agreement to pay slightly higher rents in order to build up a fund used to later purchase and expand their pool of housing, (which later increases the benefit they receive from economies of scale, and reduces risk of a major issue in one building causing a lack of revenue altogether) or a new cooperative is formed with money pooled from members, and once a specific threshold is met, they collectively purchase the property.

Housing is a good with inelastic demand. Everyone needs housing. There will always be someone, or some group of people willing to buy. And if you don’t have landlords to artificially increase the price of housing, which only goes up so quickly because of its commodification, and further purchasing by for-profit landlords, then the overall cost for a cooperative to outright buy a new property, or for a new cooperative to raise the funds required, is substantially lowered.