The 15-minute figure is what they currently designed for because they have mass constraints. If they wanted a different figure (like forever) they could do it with a larger radiator.
As you said, the physics is solved! We know exactly how much heat a surface radiates in space--there is literally an equation for it. We know how to cool stuff in space. Will the price be low enough to make a profit? That's the real question. But stop worrying about cooling in space.
If you have something useful to contribute regarding how to actually reasonably put up enough mass to make this remotely a feasible idea then please contribute. We have the technology to prevent global warming, and doing that is far easier than this, and yet even that is apparently not feasible for humanity.
What's unfeasible about that? SpaceX has already launched 10,000 Starlink satellites. Falcon 9 launches 150 times per year.