It is. In fact, the general economic concept of rent and rent-seeking are generalizations of the inherent features of real property rents, which are the source of the name.
(EDIT: Fixed egregious touch keyboard vomit, clarified “land” to “real property”.)
No supply is created by person A buying a rental building from person B. Person A paying a higher purchase price does nothing good to the rents charged in that building.
Yes, there is no lobbying or current manipulation of the government, but for small landlords it's often just a game of "I want to take advantage of population increasing over time to make more money over time" coupled with "I want to take advantage of other people improving the economy so that I can charge higher rents as incomes increase." Neither of those are productive, and the latter is hard to avoid calling parasitic. They don't have the ability to redevelop into higher density, or often even do much to change the condition of the property. So you get small mom-and-pop rentals in SF or similar places that charge far higher prices for far shittier units compared to those put on the market from larger developers or management companies in, say, Marietta, GA.
In a free market, pushing up prices of apartment buildings would also cause more apartment buildings to be built. Of course many places like SF are not a free market because NIMBYs block most construction, which keeps supply down and prices up.
And no, pushing up the prices of rents don't incentivize building. Building rentals is meh compared to just buying existing housing supply. While house prices are increasing, so will rental prices commensurately, because renting and buying a house are not isolated and independent choices.
In reality, there is no increase in construction, because the increase in housing demand is no different from someone buying a house, and the increase in property costs that follows it doesn't change anything.
They're two separate markets. Increasing rental supply does drive down rents.
If in one location, rents are very low but house prices are very high, then landlords would have incentive to take properties off the rental market in that location and sell them, perhaps using the proceeds to buy investments elsewhere.
Increasing owned housing supply decreases property costs. Decreasing property costs reduce the costs of renting, and they also make buying a house more competitive than renting, which is why increasing housing supply, not rental supply, drives down rent.
Beyond that, increasing rental supply alone will not decrease rent. It's perfectly viable for rents to increase despite more rental units if the alternative of buying a house got more expensive. See the recent increase in rents as property prices increased, despite no change in rental supply.
I actually think this case is worse for the overall economy than the rental building scenario!
Perpetual rentership is more economically precarious and less stable than ownership. But here we have the folks with assets and income sufficient to purchase properties buying up additional ones to pull ownership further out of reach for more people. So yay, we've converted a property that someone might've otherwise purchased as an asset to yet another rental.
I also don't agree with your claim that in a free market this would cause (much more rapid) increased construction in most cities, given current attitudes about small-landlordship. You can't build there without tearing something else down first, and small landlords are not often going about replacing ten unit buildings with twenty unit buildings! That requires a different level of capital. In SoCal can see this in neighborhoods in places where the current zoning allows, say, 5 story buildings, but those new five story buildings are still sparsely scattered amongst old two-story buildings that are simply too attractive to keep (the dream people are pitched that makes them become small landlords isn't "sell everything eventually to a developer, it's "passive income".) Small landlords made a lot of fuss about earthquake retrofitting, a much smaller expense than "kick out all my current income-generating tenants and rebuild it all." It's been much easier for developers to convert old low-density commercial or industrial property, since the market is more fluid there (and often with much more slack in it, in large cities).
I similarly don't expect a high number of single family lots to turn into four-plexes in the next fifteen years now that CA has changed zoning on them. It's definitely different at four units than two - where duplexes aren't selling for any more than SFH currently, since the qualitative demand for a true single family home is just that much higher - but I don't think it's enough different for a significant number of people to want to cash out or try to play developer.
(NIMBY's exist everywhere, too - acting like they're unique to big expensive cities is self-defeating. The way lower-cost-of-living cities have maintained those costs is a combination of less demand and more land to expand into. If you can build your new mega apartment complex on empty land, you don't have to convince those small-time landlords or property owners to sell all their separate buildings! Dallas County didn't increase in population much more rapidly than SF County because of rapid increases in density - just look at the Dallas city population vs the burbs.)
That's not the economic definition of rent seeking.
> and landlords riding along a wave of demand that they help create
Demand for owning, yes. But for renting they create supply, not demand, in effect helping lower rents.