You don't need to gain control of the assets. You just sell some of the stock for a profit as it rises (and only getting capital gains tax rates). This can be repeated ad infinitum assuming the underlying asset value grows indefinitely and the stock keeps splitting.

But someone, ultimately, must be able to. You can't exchange your 1% of shares against 1% of the company assets, unless you have enough share to control the company.

Let's take Facebook, where I think Mark Zuckerberg has ~60% of the shares, and only 40% are on the market, thus even if you buy them all you cannot gain control of the company. You can't extract value from the company from this avenue.

Again, what gives inherent value to a stock is that you, or someone else, will be able at some point to get value from the company itself, not from someone else. In a rational market, a price rise should mean the market will eventually be able to extract more value from the company then thought.

You missed lending the stock to others for an interest payment, and when there exists other market participants with a view the stock will be worth more than it currently is.

I don't think it ultimately gives inherent value to the stock, because you don't extract value from the company with this mechanism, but from others.

Let's take a planet in another galaxy. I would argue the inherent economic value of this planet for a human on Earth is null. You cannot and there's no realistic time frame in which you would be able to extract value from owning it. And yet you could sell the ownership, lend it, ..., and make money. That doesn't give inherent value to the planet, it's just speculation.