Does a check on the judiciary preclude the notion of an independent judiciary?  Perhaps.  It seems to be a moot point, however, which means, not a settled point, but one that is debatable.  My idea is that if judicial independence means no interference whatsoever, that invites tyranny, if it is not by its very nature tyrannical.  If independence is to be maintained, then no governmental agency, entity, or person should be allowed to intervene, and it seems eminently logical and practical and patriotic, therefore, to provide for intervention ONLY by the people, as a DUTY, and as a means of participating directly in the political process.  Someone once said that "a mistake is an error left uncorrected."  We're just correcting an error, made perhaps by the founders or perhaps by interpreters with a personal interest in immunizing judicial officers, before it becomes a more serious and perhaps a mistake fatal to the Republic.

I agree with your correct presumption, by the way, concerning the nature of the establishment of government.  Webster's definition of Civil Liberty went on to explain that:

"Civil Liberty is an exemption of the arbitrary will of others, which exemption is secured by establishing laws, which restrain every man from injuring or controlling another.  Hence the restraints of law are essential to civil liberty.

The liberty of one depends not so much on the removal of all restraint from him, as on the due restraint upon the liberty of others.   Ames.

In this sentence, the latter word liberty denotes natural liberty."

No one would argue that to prevent injury or control laws should exist.  But when the restraints become excessive or arbitrary, as they have, so that they are NOT "necessary or expedient for the safety and interest" of the people, but for the government itself and for the corporations it creates, --that, Webster says, is "tyranny and oppression."  Ah, there's the rub.  Now judges uphold bad laws, badly written, ill-conceived, and misapplied.  Their only defense is the Neuremberg defense:  They're only following orders.  And they are, not as judicial officers, but as employees of the State exercising Executive power.  In Thompson v Smith, the supreme court ruled that "judges, when enforcing statutes, do not act judicially."  WOW, ya gotta love it!  That's because you must have Laws, Statutes AND Regulations for the due process of the law to be complete!  Otherwise, the "judicial" process is flawed, defective and/or abusive.

The only way for a judge to be a judge is to be delegated authority by Congress through Article III of the National Constitution.   It says Congress MAY delegate judicial authority.  In fact, the record indicates that Congress actually withheld the delegation of judicial authority!  Check out Northern Pipeline v. Marathon Pipeline, a bankruptcy case in which the court declared that a bankruptcy court judge does NOT have Article III judicial power, is NOT a judge, and so his decision was null and void.  Ouch!  Who woulda thunk it? 

What would happen if the same were considered about State Judges?  I know what the ruling SHOULD be.  They're not Article III judges either and State courts are NOT Constitutional courts.  That definition is, not of a court that considers, uses, abides by and is restricted by the constitution, but one that is a court established or created by the constitution.   State courts are created by the legislature.  State judges, therefore, are NOT judicial officers, and are not independent, but have conflicts of interest inherent in their job!  What does it say on their paycheck about who pays them?  They take a loyalty oath to the State.  The State owns the judges, the court, all the attorneys, and they are the plaintiff trying their own cause.  How the hell did that happen?  True or not, I even heard that judges LEASE their courtrooms!  In any case, it is merely a "kangaroo court" as I said before,  "one in which the outcome is pre-determined."  And so is the income!  They've got a cash cow going and they don't want to give it up.  A portion of the court fees go into their retirement funds.  That's a financial interest in every case.

So it is up to the people.  As much as I hate cliches, it is apparent that the ball is in our court, pun intended.  It is, and must truly be, not the State's court, but the people's court, or it is no court at all.