You are confusing engineering challenges with show stoppers. Cooling in space is a well studied problem with a few possible solutions. They all boil down to needing a lot of mass to radiate heat out to the universe and ways to conduct heat. We've been doing that at small scale for decades. SpaceX is already operating a fleet of many thousands of satellites that they built and engineered. They'd be well familiar with this challenge.
Once you have solutions, it turns into a cost problem. And if that cost is too high (for whatever arbitrary threshold you use for that) it becomes an optimization problem.
This whole thread reads like a lot of "but ... but ... but ...". It all boils down to people assuming things about what is too much or too hard. And it's all meaningless unless you actually bother to articulate those assumptions. What exactly is too hard here? What would it take to address those issues? What would the cost be? Put some numbers on it. There are also all sorts of assumptions about what is valuable and what isn't. You can't say something is too hard or too costly without making assertions about what is worth paying for and what isn't.
The answers are going to be boring. We need X amounts of giga tons launched to orbit at Y amount of dollars. OK great. What happens if launch cost drops by 1 or 2 orders of magnitude? What happens if the amount of mass needed drops because of some engineering innovation? Massively dropping launch cost is roughly what SpaceX is proposing to do with Star Ship. Is it still "too hard"? You can't have that debate until you put numbers on your assertions.
There's a bit of back of the envelope math involved here but we're roughly talking about a million satellites. In the order of ~2.5 million tonnes of mass (at 2.5 ton per satellite). Tens of thousands of Star Ship launches basically. It's definitely a big project. We're talking about 1-2 order magnitude increase of the scale of operations for SpaceX going from lower hundreds to thousands of launches per year spread over maybe 10-15 years to work up to a million satellites.
I'm more worried about what all that mass is going to do when it burns up in the atmosphere / drops in the oceans. At that scale it's no longer just a drop in the ocean.