> They'll pay tax on that realized income. And if they build up all that wealth in stock value, then if we don't tax it while they are alive we'll take it when they die, so we'd take away the incentive to hoard all the wealth with those schemes.

I don't think you quite understood the scheme here: they never own the PMC. They don't gain the appreciation value of the house or anything like that. It's not an investment to them. They just got someone else to set up the PMC and buy the house, for the positive side-benefits that come from having arranged to assume (at least partial) directorship over that PMC.

Maybe it's easier to picture it at larger scale, with an explicit example: say you're a famous property developer who has built a bunch of hotels with your name on them. In one sense, those are businesses for their own sake. But in another sense, those are your vacation homes. You can stay in them, for free, any time you like, which the hotel will likely book as a business expense (hosting one of its directors for an official evaluation of operations on behalf of shareholders.)

That business expense will also extend to any perks the hotel offers its ultra-premium clients (often a class only set up in the first place as a way to justify giving directors these perks) — e.g. hotel-employed drivers assigned to pick the billionaire up from the airport, and then attached to them to fetch whatever they need as long as they're there.

If you build a bunch of hotels that provide you a free place to stay anywhere you go, then you don't need vacation homes. You never need to buy them, rent them, own them in any sense. They're never realized as income, any more than a "company car" or "company-supplied laptop" that you need to give back when you quit is income.

By itself, this would be ridiculous; it'd be like starting a car rental agency just to have a "company car."

But, because these are also profitable businesses for their own sake, you can get investors (incl. your own investment businesses) to put money into them, such that you never need to put any of your own (post-tax) money into them. You don't actually even need to personally be a shareholder in the hotel chain, "only" a director of it, to reap these benefits! And, if you're also a person with a lot of connections — someone with a rolodex of reliable CEOs and executives to staff these businesses with — then you never need to worry about actually building or running the business, either.

This is how billionaires — or at least, capitalist billionaires — get away with never realizing much income (or even having very many estate-tax-able assets as part of their net worth!) at all. They just cause corporations to invest money in other corporations, never actually touching any of the money or the resulting assets themselves, but rather ending up in arrangements where business expenses of the invested-in corporations are being spent directly on their own happiness, mostly by having temporary use of the business's assets whenever they like, as some non-transactional, pure-expense part of the business's operation. The vast majority of their assets are soft power — social influence — realized/"invested" by putting people into power in roles in businesses, where those people in power in those roles can then give them perks in return.

(And, if they ever need real liquid assets for some reason, then they just make arrangements with other billionaires to 1. start non-profits; 2. get some underemployed accomplices [e.g. relatives] employed at their non-profits in high-paying [but not to the point of being individually highly taxed] roles; and 3. get corporations that owe each billionaire favors, to direct charitable donations to one-another's nonprofits. Then, since this income is entirely the result of cronyism, the accomplices will listen when the billionaire directs them on how to spend their income — and thus the billionaire can spend it as if it were their own income, as they please; usually not on assets, mind you, but rather usually for expenses that personally benefit the billionaire, e.g. things like huge parties or political lobbying. This is the classic "high-society wife" setup, but it can be extended to hundreds of accomplices.)