Note that, according to marginalism, profit vanishes at equilibrium and capitalists, on average, earn only interest on their capital.

I don't think that's true, at least once you go beyond simplified intro-to-economics kind of texts. In a very very crude model (which is not specifically marginalist) with no frictions capital costlessly and instantly flows between different applications equalizing their risk-adjusted rate of return. But no one pretends this crude model is anywhere close to reality.

Also, you're not making any predictions. You're making an observation. Plus, of course, the equilibrium is an unattainable state, an attractor for chaotic motion. While you are drawn to it, you never actually get there.

And yes, I would argue that demand does not, in the medium to long-run, influence "value" or "long-run average market price."

That... depends on the context. In a very simplified scenario sure. In a bit more complicated scenario which includes, say, the economies of scale, it does. Consider e.g. the "value" of a widget the demand for which is a couple of dozens a year. It's likely to be very expensive (= high "value") since at such levels of production it's going to be hand-assembled. Now let's say it got popular and you can sell a couple of millions a year. This makes it worthwhile to build a proper assembly line and the widget magically gets much, much cheaper (= low "value"), all because of demand.

Your example of oil is also a bit more complicated. Oil does not have a uniform price of production. Each well has its own. This means that when demand is low, high-cost wells are shut down (and the low-cost wells produce cheap oil), while when demand is high, high-cost wells are drilled and/or reactivated producing expensive oil. In this way demand directly affects the global price of oil. When it's high, low-cost wells make a lot of profit, when it's low, high-cost wells are turned off.

Marx's labor theory of value

You do realize there are reasons why it's not all that popular nowadays? In particular, the notion of value as labour crystallised in the product has a lot of issues. Even Marx himself realized the problems by the third volume of Das Kapital.

But, in any case, you're saying it explains "long-run changes in prices of production and the average world rate of profit"? How does it do that? What does it say will happen during the next couple of decades?