Moreover, you can easily calculate how big of a radiator you need for a given power level and temperature. You can use the Stefan–Boltzmann law:
A ~ (1000 P) / (2 e k T^4)
Where A is the radiator area in square meters
P is the power in kilowatts
e is emissivity (usually 0.9)
k is the constant 5.67e-8
For a 1 kW test like Googles, you just need 1 square meter of radiators (assuming two-sided).For SpaceX's 175 kW satellites, they will need ~170 square meters of radiators.
None of this is impossible or even difficult to calculate. That's why I think this is the laziest argument against space data centers. There are so many other more reasonable arguments (like whether they will be economically competitive) but people love to latch on to this one for some reason.
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?