Platinum group metals would be very valuable on Earth, but it would be very expensive to purify them in space, move back to Earth and deorbit. Perhaps it will make money but I am skeptical.
Aiming for it looks like a much simpler objective than crewed Mars missions, with vastly greater potential for kick starting something profitable, ie self supporting. Vastly simpler and smaller scale than attempts at colonization - which must manage the costs and difficulties of mining and refining in extreme environments too, with far more at stake and no potential for profitability.
Very big ships (Starship) with reduced costs for launching equipment to space (unlikely to require many) would help but the only ways I can see that asteroid mining might work would require high durability solar electric propulsion once everything is in place, using fuel/reaction mass sourced entirely from asteroid materials thereafter. Sort of like the imagined independence from Earth, as in must make do with absolute minimal ongoing materials from Earth, but much easier by not including people.
Unlikely to get in-situ refining to pure PGM's, let alone to individual elements - but the
Mond process appears suitable to reducing nickel-iron to nickel plus iron and leave a high value residue high in PGMs. Nickel refineries do that. More cobalt than PGM's going by meteorite samples. $Millions per ton rather than 100's of millions per ton? The rocketry - as for most space ambitions - is critical, but not likely to be Starships.
I agree it would be extremely challenging to attempt to mine (preferably) near earth asteroids and make it work but pilot programs to trial essential elements - a suitable rocket vehicle for in-space transport, mining and refining "fuel" to run it, developing and trialing a zero gee suitable Mond refiinery - all far less difficult than Mars missions let alone colonies and not using crews, so no lives at risk.