Diamonds aren't dense, not especially.. They are the hardest naturally occuring substance and only beaten by a few very odd man made compounds.
The expense depends on size, disproportionatly so. Industrial diamonds (fraction of a carat sized) are indeed dirt cheap and fairly common. As you get bigger you get rarer and hugely more expensive.
A typical mine will 'lose' 80-90% of the total carat value in processing, but that only accounts for less than 1% of the money value. That's how 'worthless' small carat stones are.
On the other hand if it weren't for jewelry the large diamonds would be equally worthless, they have no inherent value, just the rarity and sparklyness of them. Which is part of the reason man made jewelry diamond haven't caught on, a man made large diamond is like a knock off designer clothing. It's a statement of wealth and status, so buying a cheap one defeats the point (unless you haven't got the money, but want people to think you have
)