In supply-constrained markets, wouldn't any benefit from this eventually accrue to landlords rather than renters? Put another way, if landlords are charging what the market can bear, and the market can suddenly bear 25% more, won't they just raise prices over a few years to eat up that new wealth?

You are correct, all of this benefit would end up being taken by landlords due to their ability to dictate prices - btw government subsidized housing can be a different situation and the government can be an intentionally inefficient market actor to benefit the renters. I think there are still more accurate way to subsidize the lower class though, a reduced income tax is more likely to directly translate into a higher savings rate and more consumer power - and that last bit benefits us all by increasing the velocity of money.

Would be an incentive for the city or state to enact policies for making housing more affordable. From what I've seen a lot of housing issues are coming from not-in-my-backyard-ism and zoning.

I disagree, we shouldn't subsidize renting as it encourages a bad economic behavior. But! Any deductions or tax credits for home ownership/property taxes should be removed.

Some of the specialized property tax reductions (like urban agriculture) can stay, but ideally we'll be pretty selective and get rid of the silly religious property tax exemptions.

Huh? Renting is bad economic behavior?

What's your alternative? Become a slave for 30 years to maybe afford to buy a house?

If you could deduce rent (or mortgage for that matter) then more money would stay in the hands of ordinary people. It would also encourage cities to make housing affordable. If housing costs get too high then tax revenue would drop. I think that's a good incentive overall.

The person renting out property is exploiting a market inefficiency to the loss of the renter. The cost of renting property is, by definition, slightly higher than the cost of getting a mortgage on the same property and gaining value in the property, otherwise the renter has no motivation to enter into a contract, there is a bit of fuzziness in that owners are responsible for the lion's share of the maintenance but in general we can assume most owners price their property so that they can cover maintenance comfortably. The only thing that drives the cost of renting below the cost of mortgage + maintenance is the fact that the owner is investing their time owning the asset in subsidizing the price of having purchased the asset - a renter is always going to pay more money (barring crazy stuff like living in an area with frequent natural disasters, but the market will adjust to that)

It is a good life choice to try and recoup some of the money you're paying into housing as an accrued asset, if you're currently renting I'd suggest looking at the numbers in greater detail - if you move frequently the tax your government levies on selling properties may tip this equation the other way but generally property ownership is a good idea.

To return to your original question, a subsidy on renting allows property owners to shift the price point on their rentals in their favor and ends up going into the owner's pockets, they can reinvest and grow their assets faster while people who are in the rental market will find it harder to escape due to an increased growth in property values - also a rental subsidy is great for anyone who already owns but I don't really think that's a section of the population we need to actively subsidize.

Keep in mind that not everywhere has Bay Area real estate prices.

I bought my place for 150k back in 2002. I paid it off in less than 7 years, and have been mortgage free for over a decade now. I know another guy who did something similar, paying off his place is less than 6 years.

I realize this is very uncommon. Most people I know are perpetually refi-ing, and have no desire to ever pay off their mortgage.