It would be reasonable to expect some level of concern from the governments of Earth that even relatively small metallic objects could be used as effective weapons if lobbed at specific cities or military installations. Perhaps heavy manufacturing will for that and other reasons be kept away from Earth orbit and instead be done in, say, lunar orbit or in one of the Earth-sun Lagrangian points where the gravity well isn't so deep and it's easier to move the product anywhere else in the solar system.
Have you seen the reports on the new data coming out of the mission orbiting Bennu a few days ago? It's looking like small asteroids might be a bit less undifferentiated than we thought those the story isn't changing by that much.
https://www.space.com/asteroid-bennu-formation-history-osiri...
In retrospect I guess it makes sense that if 16 Psyche is a chunk of planetary core, then the sort of impact that created it would likely have turned it molten, and its internal weak gravity may then have caused all the heavy metals to sink to the middle before it cools.
At any rate, gold mining sounds like an easier problem than, say, trying to manufacture stainless steel in space in a way that makes economic sense. (We'll probably get there eventually if we're going to have any kind of real space economy, but bootstrapping is going to be hard.)
That's not how it works.
Simply put, most small asteroids are undifferentiated. Meaning, they all have roughly similar mix of materials that depend mostly on the distance to the sun of where they formed. Unlike on earth, since they formed they have never been molten or been subject to erosion and transport by weather and water, which concentrates like materials together. Most asteroids are basically balls of dust where any grain is pretty much in the place where it landed when it first hit the ball.
The upside of this is that gold (and similar heavy metals) is much more abundant in asteroids than it is in the earth's crust, because when the whole earth was molten, all of our native gold ended up deep in the core. The gold we do have on earth is mostly what has rained down in meteors since the earth's crust has been solid. If they were on earth, each and every asteroid would be made of an exceptionally rich gold ore. However, add the cost of moving all of it back, and even with implausibly good rockets, it's just not worth it.
The feasible options are either:
1. Refine in situ. Develop some process of separating all that gold (and other valuable materials that are much more abundant in asteroids than on earth) from the less valuable materials, and then just send the gold back. The fact that we are talking about dealing with a ball of barely-compacted dust makes this in some ways easier, in others harder.
2. Find one of the much rarer differentiated chunks of rock and metal instead. They have much higher concentrations of the stuff you want, and are in many ways much more convenient to deal with, given how they actually have a hard surface and all. The biggest problem with them is that they are generally going to be very big. As in, less rocks floating in space and more minor planets. The most promising candidate for this is 16 Psyche, which is believed to be an exposed iron core of a protoplanet that got smashed apart by a very energetic collision. It probably has more gold than all of earth's crust, and it probably exists as an uniform solid gold layer.
The problem with 16 Psyche is that you are not moving it anywhere. It's >250km across and masses more than 2 quadrillion tonnes. So you have to dig into it. And, the layers above the gold layer are made of solid nickel-iron. So you either hope that there is a crack into the deep layers formed when the protoplanet was busted apart, and use that, or you somehow tunnel through a hundred kilometers of iron.