Not "cost of production," but "price of production," which includes the cost of production plus an average rate of profit.

Note that, according to marginalism, profit vanishes at equilibrium and capitalists, on average, earn only interest on their capital. I disagree. At equilibrium (over the long-run), an active capitalist (someone who employs capital to produce commodities) can expect, on average, to make a rate of profit that is at all times strictly above the going interest rate. The average rate of profit must always include some substantial amount of "profit of enterprise" to account for the added risk of producing and marketing an uncertain product rather than just being a financial capitalist and earning an interest rate (which carries a typically lesser risk of the default of the debtor). If the rate of profit is not substantially above the rate of interest, over the long run you will see capitalists transition from productive investment into financial capitalists (a problem we have right now). This will eventually decrease the supply of commodities and increase their relative prices until it is once again profitable to produce commodities even after deducting interest.

So, that is one concrete prediction. And empirically, although the averate world rate of profit can sometimes briefly dip negative, it is as a rule on average a substantial positive percentage.

And yes, I would argue that demand does not, in the medium to long-run, influence "value" or "long-run average market price." It's all about the price of production instead. The practical advantage of this is that this opens up opportunities for arbitrage against other people who don't realize this. For example, if it is 2007 and you see demand for oil surging and the price skyrocketing, you should keep in mind that the price of production of oil has probably not changed that much (excepting the fact that some of the new oil being brought online was shale oil that had a higher price of production), and thus oil will be making above-average profits at these prices. You can then expect investment to flood into oil production over the next ~5 years, thus increasing supply and bringing the market price of oil down to (or even temporarily below) the price of production. If I had had some money back then instead of being in high school, and if I had known what I know now, I am confident that I could have made some serious money on some sort of long-term oil future betting. Note that, now that I do have a little bit of money, I am indeed making plays in the market right now based on my analysis, although I won't go into specifics about what those are right here...

Note that, so far, we have been taking the price of production of various things and the average rate of profit as readily-discernable "givens" at any point in time. However, prices of production and the average world rate of profit can change as well over the long term.

Even the classical economists didn't really have a theory for what determined these changes. (For example, Adam Smith could tell you that the cost of production was the rent + capital + wages that, on average, was needed to produce something, but how can you anticipate changes in the costs of each of those? And then you need to add on the average rate of profit, but how can you anticipate how the average rate of profit of the world economy will evolve?)

So far, the only theory that I have seen that even tries to explain long-run changes in prices of production and the average world rate of profit is Marx's labor theory of value. For example, see: https://critiqueofcrisistheory.wordpress.com/responses-to-readers-austrian-economics-versus-marxism/why-prices-rise-above-labor-values-during-a-boom/

Note that you don't have to buy into Marx's labor theory of value to do medium-run arbitrage involving prices of production. All you need is classical economics for that.

Only if you wanted to do very long-run arbitrage that took into account technological change and the resulting increases in the productivity of labor in certain sectors—and thus declining production prices for those commodities and a long-term tendency for the worldwide average rate of profit to fall as the so-called "organic composition of capital" increases—then you would have to rely on Marx's labor theory of value or some other yet-to-be invented theory that could attempt to forecast changes in prices of production and the average worldwide rate of profit.