What do you mean by "Earth-born conspiracies?" If I remember right, the initial mission for Stormwatch was to protect the Earth from massive extra-terrestrial threats and keep them in the dark.
First five aren't extra-terrestrial, Martian Manhunter appeared in Legion Lost and GLC, and who knows, maybe at some point a member of Stormwatch did check out a certain crash landing in Kansas, figured it wasn't an real or immediate threat, and helped keep the government off Ma and Pa Kent's back.
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The Firestorm projects
The Rot
several of the Olympian Gods were hanging around in human cities.
And besides conspiracies:
Did they happen to notice that virus from the future in Legion Lost?
How many aliens crashing to Earth have they caught notice of? A Kryptonian, a Green Lantern
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