Putting aside the question of from whom are the judiciary independent, the current structure of the three branches, in relationship to the democratic masses, is I think the main objection. The Supremes serve for life ("during good behavior"), are appointed by President, confirmed by Senate. The Senate has already been mentioned, but important also are the long terms they serve. Were every part of the three branches directly elected, with short terms, and other structural deficiencies fixed, there would not be much gain to democracy, in my opinion. The first-past-the-post rule coupled with the freedom of Money to enter politics (not to mention the lack of an independent national media) is the real killer.
But since you mentioned an "independent judiciary", I couldn't resist digging up a tidbit on Rehnquist I wrote to a few friends of mine last May:
Rehnquist is an arch-conservative (in real terms, a reasonably well-mannered fascist) who wrote a memo early in his career that Plessy v. Ferguson (1896) "was right and should be re-affirmed." Plessy, remember, was the case that *upheld* the "separate but equal" bullshit allowing de-facto segregation of education. Rehnquist also authored another memo to his then-boss, Supreme Court Justice Robert Jackson in which he urged Jackson to vote against a black challenge to all-white Democratic primary elections in Texas, saying "it is about time the Court faced the fact that the white people in the South don't like the colored people". Further burnishing his reputation as a judicious thinker, Rehnquist in 1974 (three years after being appointed to the Supreme Court) purchased a home with a covenant which prohibited ownership by anyone of the "Hebrew race". Guess he forgot his spectacles that day!
I think I got most of the above from one of Philip Burch's volumes...
Bill