Then tax land. Make landlords sweat if they fail to rent out units.

The solution to housing affordability is all about negotiation power. Currently, tenants in many cities do not have negotiating power because an overall shortage of housing and thus no credible threat of vacancy (and with no land tax, vacancy isn't enough of a financial loss). Build more housing by making it legal and easy to build housing, tax land, and tenants will have far more negotiating power.

A good example is my city, Montreal. Thanks to being one of the most YIMBY cities in North America, we have far more missing middle housing than any other city of this size on the continent. The result? More power to the tenants, less to the landlords. And the result of this power? Not only is my rent waaaaay cheaper for what I get compared to my sister in Boston (we pay about the same, but she has 3 roommates and lives farther out, while I have 1-br and live right next to downtown in an insanely walkable neighborhood), but I also negotiated down my rent when moving in. The reason I could negotiate down my rent was because the landlord had a credible threat of vacancy, so they'd rather lower rent by a hundred bucks than risk vacancy.

taxing the shit out of third homes for example

There's actually a far more elegant type of tax that would have the effect you desire: land value tax.

In short, it's a progressive, difficult-to-evade, and extremely economically efficient tax that -- both in economic theory and in observed practice -- cannot be passed on to tenants. Further, it strongly incentivizes new housing development, heavily penalizes real estate and land speculation, and improves affordability. In fact, it's so well-regarded a tax that it's been referred to as the "perfect tax", and is supported by economists of all ideological stripes, from free-market libertarians like Milton Friedman -- who famously described it as the "least bad tax" -- to social democrats and Keynesians like Joseph Stiglitz. It's simply a really good policy that I don't think is talked about nearly enough.

Even a quite milquetoast land value tax, such as in the Australian Capital Territory, has been shown to reduce speculation and improve affordability:

It reveals that much of the anticipated future tax obligations appear to have been already capitalised into lower land prices. Additionally, the tax transition may have also deterred speculative buyers from the housing market, adding even further to the recent pattern of low and stable property prices in the Territory. Because of the price effect of the land tax, a typical new home buyer in the Territory will save between $1,000 and $2,200 per year on mortgage repayments.

Letting the market decide the rent will not magically create more housing

Rent control does not create new housing either. The reason economists basically all say rent control is bad policy is because it drastically reduces the amount of housing that gets built.

If you want more housing, make it legal and easy to build new housing like Japan did. Tokyo is the most populous city in the world, and yet it's remarkably affordable because it's very easy and streamlined to build new housing.

In Tokyo, by contrast, there is little public or subsidized housing. Instead, the government has focused on making it easy for developers to build. A national zoning law, for example, sharply limits the ability of local governments to impede development. Instead of allowing the people who live in a neighborhood to prevent others from living there, Japan has shifted decision-making to the representatives of the entire population, allowing a better balance between the interests of current residents and of everyone who might live in that place. Small apartment buildings can be built almost anywhere, and larger structures are allowed on a vast majority of urban land. Even in areas designated for offices, homes are permitted. After Tokyo’s office market crashed in the 1990s, developers started building apartments on land they had purchased for office buildings.

https://www.nytimes.com/2023/09/11/opinion/editorials/tokyo-housing.html

Also, you appear to have a very uninformed view of who economists are and what they value. Economists, like all other social scientists, are academics and scientists. Believe it or not, economists are not a cabal of landlords wanting prices to go up; just about every economist will tell you the dangers of inflation and rent-seeking. You might be surprised to learn the surveyed beliefs of economists:

• The majority of surveyed American economists vote for Democrats instead of Republicans
• The majority are in favor of environmental protection regulations (EPA)
• The majority are in favor of food and drug safety regulations (FDA)
• The majority are in favor of occupational health and safety regulations (OSHA)

Further, the largest-ever survey on economists' views on the climate show an overwhelming economic consensus on climate change:

We conducted a large-sample global survey on climate economics, which we sent to all economists who have published climate-related research in the field’s highest-ranked academic journals; 738 responded. To our knowledge, this is the largest-ever expert survey on the economics of climate change. The results show an overwhelming consensus that the costs of inaction on climate change are higher than the costs of action, and that immediate, aggressive emissions reductions are economically desirable.

Believing economists at large to be just a cabal of greedy capitalists is just as anti-intellectual as MAGA people believing climate scientists to be a cabal of Soros- and Gore-funded fraudsters.