Like the Judiciary is independent of the Executive and provides a check and balance to its power?

Actually, yes. The judiciary does provide a check to executive power - just not as often as the press tells you it should.

I didn't buy that in grade school civics and I buy it even less today.

No enforcement authority = no authority. It's really that simple. Strongly worded letters are pointless in a world where someone can hold them up, shout "Complete Exoneration!", and have their media apparatus repeat it until their followers believe it.

Theoretically yes, but all except one are Trump appointees and the main thing he looks for is loyalty. In addition, the threat of getting publicly hammered and humiliated by the president is often enough to make someone think twice before not doing what he asks.