Let's do math.

Let's say the "space datacenter" peaks at 500 megawatts, seeing how Earth ones apparently peak around this. But that includes stuff like waste heat from power generation, the cooling and comms system, eveything on the spacecraft.

Lets say we want the coolant at 60C, so the computing stuff stays under 80C, as I am trying to give this system the benefit of the doubt. And lets assume the radiator is quite efficient and ignore mere engineering concerns, and give it an overall emissivity of 0.8.

From Project Rho:

Radiator area = P / (ε * σ * T^4)

Radiator area = (10 ^ 8 W) / ((0.8 emissivity) * (5.670374419 *10^-8, boltzman constant) * (333 Kelvin ^ 4, the same as 60C))

...That's a radiator two thirds of a mile across.

Let's, again, toss practicality out the window and say the "weight" of the whole thing is similar to a 6 mm aluminum panel, which seems like an unreasonable feat of engineering. After all, we gotta pump liquid through the thing, and unfold it somehow. But lets go with it.

That's 5400 cubic meters of aluminum. That's 5.6 * 10^6 kilograms. Picture a cargo ship flattened into a disk; that's the order of mass we're talking about.

At 20,000 kg per flight... that's 112 Falcon Heavy flights to low earth orbit, or ~$10 billion dollars. Just to get our impossible radiator into orbit, and nothing else. Lets say launch costs get 10X cheaper, somehow, and that's still a billion dollars.

Is a huge heat shield with some aerogel or something behind it to contain the heat that couldn’t be turned into energy

The first line of the Project Rho page:

NO, for the millionth time you CANNOT get rid of the heat by turning it into electricity!

And yes, we're assuming the radiator is way far away on some kind of tether. Again, first few lines of the page. But the aerogel wouldn't make a difference, it would actually hinder this system.

I will bet money that at some point someone argued that communications going through satellites in space would be unpractical because of engineering and technological challenges, the stupendous cost of getting stuff up there, and monumental maintenance costs compared to just having some lines going through some street poles.

No.

No one was saying this.

People complained about the engineering concerns, but engineers recognized it was physically practical... in theory.

But we are talking about physics now. Doesn't matter if you're a transcendental AI or not, you cannot engineer your way around thermodynamics, and we are talking about just one problematic system.

The only tech that would make such a dramatic difference (that's on the horizon) is a space elevator, as it circumvents the rocket equation entirely by 'pushing' against the Earth. But this is really, really, really hard.