"Since it’s a payment for nothing that didn’t already exist, it’s a deadweight cost, and so not only unjust but also dysfunctional for the economy. "
The practice of renting exists as an option to owning. By renting, renters can engage in an activity with less up front capital to buying. For example, a contractor building a concrete sidewalk may choose to rent the mixer rather than owning it, so that it's cost is a cost of sale for the service, rather than a fixed cost that requires maintenance, storage and transportation. In that sense, the rental is a component of a larger transaction involving actual work being performed, and not parasitic. Moreover, by offering the mixer for rent, the 'rentier' reduces the total number of mixers that need to be produced, as it is a shared resource to contractors and they each do not need to own their own.
The same principle pertains to business. A fast growing startup may know they only need a space for a short time and will grow out of it. So it would make no sense to purchase the property they will inhabit for a short time. The rent they pay is an alternative to mortgage or cash, and it makes sense for their business purposes, which in theory do produce something of value.
In the example of home rental, making homes available for rent enables families to live in accommodations that they would not be able to purchase. These accommodations are not without cost to the owner: Taxes, utilities, maintenance, security and administration are all associated costs that rent payments offset. What the rentier provides in most cases, is a better quality of life than the family could have if they were forced to own the property they lived in.