But having 1 entity control such a large amount of supplies means that entity can control the market rate (they simply hold on to it or sell it as needed). National banks, like the Bank of Canada, already do this with their country's currency.
Second, if it works out for them and they're able to start mining these, other people will start doing it, too.
What if the pioneer's space manufacturing, using their cheap and plentiful resources, gives them an insurmountable lead? They win the race to claim and defend new prospects against latecomers.
(To the planetbound, this might just look like a suspiciously long string of failures by all subsequent foothold missions…)
War.
However, the good news is that even on Earth the benefits of cooperation tend to outweigh the benefits of aggression, in space for the foreseeable future it will be even more so. It is so, so easy to trash a space facility and so, so hard to defend against it that in essence all private companies in space will de facto be in a MAD situation should they decide to go that route. It will be a long time before anyone will have a position where they can afford to attack somebody and have a reasonable expectation of survival.
As a private company--especially if this mineral becomes very necessary and expensive--would be a very big target for some sort of 'congressional oversight' to 'protect national interests' and prevent a harmful monopoly.
Cars aren't worth very much. Try reselling a car to find out how little they're really worth.
Shoes aren't worth very much. Try reselling a shoe to find out how little they're really worth.
Imagine they release 100 tons / year.
The world supply of platinum has just gone from 100 to 200 tons / year.
Or they release 200 tons, and the world supply goes up to 300 tons / year.
At some point really low they'll make platinum as common as dirt and worth about as much.
* exaggeration to make point. Point stands.
You cannot value the asteroid at weight * current_price_per_ton.
Because the earth is surviving on 100 tons of platinum / year, and the current_price_per_ton is based on demand to supply.
If you rocket back with a whole 100 tons in the first year, you dramatically affect supply. The price drops; platinum suddenly becomes more affordable for more things, demand may increase and so on.
But we can all see that if there were mile cube of platinum delivered to earth, it wouldn't actually be worth $1500/ounce any more.