The metals that are much more abundant on asteroids than on Earth will be dissolved in the Fe-Ni-Co metal in concentrations varying between 2 ppm for the most abundant (ruthenium) down to 0.05 ppm for the least abundant (rhenium).
While these very low concentrations are still thousands of times larger than the average concentrations on Earth, mining them on asteroids would still require processing thousands of tons of Fe-Ni-Co metal for a few kilograms of precious metals.
On asteroids that have never been melted, the processing could be easier, because most of the precious metals might be present in very small refractory grains dispersed between the grains of Fe-Ni-Co metal and silicate minerals and maybe a cheaper separation method could be found than for the case when they are in solution.
However, the same huge quantities of material need to be processed.
Right now, it is quite certain that this cannot be profitable.
Some time, in a more distant future, we can imagine a technology much more advanced than what we have now, which would enable sending some robots able to perform completely automatically the tasks of building from local materials some huge installations for energy collection, for mining and for extracting the desired elements, so that asteroid mining would require the transport in both directions, between Earth and the mined asteroid, of only very small quantities of materials and equipment.
Even if this is much beyond our current capabilities, it might become a necessity if we would exhaust the exploitable reserves for some of the least abundant elements, dispersing them in junk from which their extraction could become too costly.
On the other hand, there are numerous research projects now trying to replace the use of less abundant elements with the use of more abundant elements, in a lot of applications.
In most cases, it is likely that such substitution attempts are likely to succeed much earlier than the time when we would be able to mine those elements from outside the Earth.
A kilogram of nickel costs ~13$. And nickel makes up a mere 0.0009% of the earth's crust.
It's completely pointless to mine them, with the hope of returning them to Earth. It's also completely pointless to mine them, with the hope of using them in space, for many, many reasons.
1. A spacecraft factory employs thousands of people, and requires hundreds of millions of cubic feet of space.
2. The supply chains that feed a spacecraft factory employ hundreds of thousands of people, and require billions of cubic feet of space.
3. They also require a long tail end of chemical inputs that are not iron and nickel.
The difference between 'We have a space factory that is fed by an asteroid miner and builds more spacecraft/space factories' and 'We have a proof of concept where we spent a billion dollars to mine and refine 20 grams of iron, which we fed to a 3d-metal printer (Never mind all the other consumable inputs into it), to print a little figurine of a rocket' is rather large... And only the latter is achievable in my lifetime.
I'm not saying it won't still be big. You basically need community in Low earth orbit, low lunar orbit, or on the moon. But it could well be worth it; depending on the demand for hardware in earth orbit to begin with.
If you get moon launch costs low enough (and they'd be low!) you might start competing with some traditional industries on earth.