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Why can't those be moved to space?

Because cherenkov radiation is only observable in a transparent medium like air or water. Space doesn't work because the speed of light in space equals the speed of light in vacuum, therefore no cherenkov radiation.

Couldn't they look down?

The Cherenkov emission is boosted in the direction of travel, i.e. downward. And there are too many lights on the surface of the Earth.

Cherenkov telescopes for neutrinos do look down for reasons of schielding. Neutrinos can pass the earth, other particles not so much.

But those are a special case, air cherenkov telescopes are looking for "less weird" particles like photons or protons. Those can only be seen looking up, since the primary particles moving down focuses the cherenkov light down in a narrow cone.

While this certainly requires non trivial large scale space engineering, you should be able to build in in principle.

Basically a big bubble filled with the most useful gas for this + bunch of photodetectors inside at the appropriate places. You could also make the whole detection chamber much larger, than the ~30 km (?) of reasonably thick atmosphere you get on Earth.

They are imaging the flashes as the relativistic particles slam into our atmosphere. No atmosphere, no flash.

https://en.wikipedia.org/wiki/IACT

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