If it's just one company doing the mining, there's no reason to crash the market. Hold 100x the world's annual demand for platinum and sell off 1% per year for 100 years. 20 Billion times 100 years isn't so bad.
Not the most ethical business plan, mind you....
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.
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.