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.