No one’s mentioned the fact that, if you have the energy and delta-V to go asteroid hunting, then you also have the capability to turn those asteroids into WMDs. No government is going to lightly cede control of that capability to a private company.
As it is, I have a very difficult time imagining that launch costs will get low enough in the foreseeable future to make this endeavor profitable. While I’m writing this, I feel a bit like King Henry of Portugal giving Columbus the brush-off, but still. Even if SpaceX, etc, drop the price/kg. to LEO by a factor of 10 or 20, I can’t see it being practical. The answer, IMHO, as to the asteroids they’d be looking for would be those containing various rare earths. Maybe Helium-3?
I would love to see the capability of asteroid mining become practical though. Because if you have the capability to asteroid mine, you have the capability to: watch for and deflect earth-crossing meteoroids/asteroids, build solar power satellites, build very large astronomical equipment, build microgravity/vacuum manufacturing. All of these could elevate human global productivity.