Ah, I think you have the same misconception as Gates in the post, of confusing the concepts of "points along the curve" with the curve itself.
> Why is it that when the price drops, consumers would want to buy more and producers would necessarily want to produce less?
I would phrase it as when the price is lower, there are more consumers who would buy, and fewer producers to sell. To take the demand curve, for example, it doesn't explain one person's behavior at different prices. Rather, think of the curve as a thousand points, each representing a different person and their own maximum price at which they would buy the item. The supply curve is a thousand producers each with the minimum price they'd be willing to operate at.
> If I was a company director and the price of the commodity that I produced dropped by a lot, I would work harder to produce more of the commodity to offset the lower profit margin
Yes, this is already reflected in the supply curve. If your company is able and willing to produce that commodity at a lower price, you're already reflected on the supply curve as one of the points on the line below the current equilibrium price. You have what's called a "producer surplus" if you'd be willing to produce at a lower price than the equilibrium.
Take the supply of uber drivers for example. If the price of rides decreases, you may to some extent see drivers adjusting how much they drive, but the main effect is to remove drivers from the market who have better uses of their time, less efficient cars, etc. Similarly, if the price increases you'll see new uber drivers coming online driving their clunky SUVs or whatever.
Or, in the case of cryptocurrency, the supply follows this as well: at a higher price, less efficient miners or people in locations with higher utility prices will be able to profitably mine, and so the number of people supplying computation will increase.
> Also in the case of cryptocurrencies, it doesn't seem to work the way the graph suggests; when the price goes up, people buy more.
This, too, works with the supply and demand curves, when you understand them properly. The concept at work, here, is that changing supply or demand (i.e. think of the thousands of people that make up each curve shifting their preferences in the aggregate) represents shifts of the curve, not movements along it. Increased demand is the curve shifting up and/or to the right (it's the same thing).
So, when the price goes up, people see it in the news and have FOMO and so "demand increases", meaning the curve shifts to the right. If you look at the graph again, you'll see that means the equilibrium price increases, which is what's happening. That, in turn, drives more breathless speculation and aggregate demand increases again, pushing the curve further to the right, driving up the price some more.
Concept check: the supply curve is not shifting, here. Per my earlier paragraph, the increasing price may mean more miners come online, but that's already what the supply curve means. A shift in the supply curve would be something like a new breakthrough energy technology making everyone's utilities cheaper. That means if everyone's minimum price before was $x, they're now willing to supply computation at $x + 5, shifting the supply curve to the right and (if you check the diagram) driving down the equilibrium price (since the # of bitcoins in circulation will increase faster and the demanded fee by the miner's will be less).