Would the US Supreme Court have had the power to undo Bill Clinton's impeachment on the grounds that the charges weren't sufficient? If not, what makes this different?
It's a completely different procedure that just had the same name stuck on it by lazy lawmakers.
How so?
Well, for one thing US impeachment is something
the House does and which leads to a
trial in the Senate. And is preceded by extensive quasi-legal hearings in the House (judicial committee) as well. It's a specific procedure to try people who enjoy immunity from normal criminal proceedings.
This is a mechanism for the Governor to remove somebody from a Commission, with Senatorial approval. There was no semblance of even a kangaroo trial. It's really,
really not what the word "impeachment" was invented for (not by any American, need I add?) The only thing that vaguely resembles an impeachment is the line about this being only in case of (do I need up the exact wording? Nah) - specifically where they're as hazily defined as in the US Constitution. Which, obviously, opens the floodgates to judicial interpretation.