Housing prices will be lower without the deduction, actually helping first time buyers more since the down payment will be more achievable. Alas, eliminating the deduction will never happen.
This seems to assume a monopoly supplier charging monopoly rents, but is an untenable conclusion if housing is supplied in a competitive market. (It also neglects the fact that its not a flat subsidy, but one where the subsidy rate is positively correlated with income, so that -- within the limits of the deduction -- higher income earners get a higher subsidy rate, and thus the spread in buying power between them and lower-income earners is increased by the existence of the deduction, which even in a monopoly-supply situation would mean that the effect across the market was not "everyone gets the same houses as they would without the subsidy, just at higher prices".)
Housing is not a very competitive market. Entitlements (whoever first claims land can generally hold onto it) and policy (predominantly zoning), leads housing stock to be fairly inelastic to demand, and those supplying housing tend to be price-takers.
Housing is not land, and quite often in the United States is not provided in a way which makes maximum practical use of land, so land being fixed is pretty much irrelevant to whether housing supply is fixed.
> so the greater ability to pay from the subsidy simply increases the 'rent' earned by landowners.
If "landowners" were a single monopoly supplier, this, again, would be a reasonable conclusion. The ability to capture value from the subsidy, however, depends on the ability to provide value from housing on that land, something on which landowners naturally compete (qualitatively, even where the do not do so on some quantitative measures, such as floor space, as robustly as would occur in a truly freely competitive market because of, e.g., zoning restrictions.)
It's clearly relevant-- the fixed nature of land makes is what limits the supply of housing. Both due to the obvious situation -- we can't build housing because there's no available land for it-- as well as the more complicated case where the limited supply of land is downzoned to create wealth for landowners.
> If "landowners" were a single monopoly supplier, this, again, would be a reasonable conclusion. The ability to capture value from the subsidy, however, depends on the ability to provide value from housing on that land, something on which landowners naturally compete.
It is uncommon for them to compete. It is much more common to enact restrictive zoning laws (and other bureaucratic measures) so they do not have to.