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.
That's like a quarter of a rack of modern AI hardware. Modern AI datacenters are now in the multi-GW range.
Now the argument is, what, you can't launch that many satellites?