Could wouldn't become more expensive then diamonds. You are missing his entire point which isn't very novel but understood well.
So lets put this in another way. Suppose you are going to build a house out of these resources. Would you build it out of coal which would cost $10 a square foot of living space, or oil that would cost $11 per square foot of living space, or solar and wind that costs $15 per square foot of living space? Suppose you decided on Coal because it was the cheapest and you could get the most for your money. Suppose everyone did so driving the demand up therefore the price of coal goes up. Now suppose the cost of the coal house is $14 per square foot of living space. Your neighbor would likely build his out of oil because again, it is the most economical at the time.
Well, this drives oil up to $16 per square foot of living space and decreasing supplies with the same demand has already taken coal up some more to $17 per square foot of living space. Now I might chose solar and wind because it is cheaper at $15 per square foot of living space. As the resources deplete and cost more, demand will change because people will start switching to the more economical resources.
Did that illustrate it simple enough for you to understand? As demand goes up or stays the same and supply decreases, the prices increase. When the prices increase past the costs of other resources, people switch to the other resources halting or slowing the increasing of the pricing and the depletion of the resource. Eventually, if you run out of a resource, society will be off it long before that happens and it will not be because society is dependent on that resource unless it is some magical resource with some special property that cannot be found anywhere else and no one can ever engineer around it.