I think demand for these metals is relatively inelastic, so even supplying a relatively small amount may crash the market. If you double the supply of platinum per year, you will crash the market price.
If you have that large of a stock, you could corner the market. Undercut the competition slowly, until they all go bust. Once the mining infrastructure is left in ruin, raise prices to whatever you want. It will take a while for the competition to re-mobilize the infrastructure to compete.
Platinum is a very useful catalyst. If it were much cheaper, we could do more with it.
Aluminum was a precious metal for a few decades. Royalty used it for their best forks and spoons. Now we make airliners out of it and the aluminum producers are doing just fine.
It's unlikely we'll be making airplanes out of platinum (terrible strength to weight ratio), and it's unlikely catalysts will consume as much as is used for structural applications (the global market for catalysts is itself limited, and it's not like we're going to be using orders of magnitude more catalysts of all types if the price of platinum reduces). Even still, the worldwide market for primary aluminum production is just $100 billion per year. And aluminum is a common ore. Even in space, platinum is rare.
Space mining is made out to be the pie-in-the-sky paydirt of space dreams, but it's still a much smaller market than telecoms.
