Both separation of powers and checks and balances are important and in a way these two concepts contradict each other. The judiciary has to be independent to check the executive, but it must itself be balanced.
There is no country with true or complete judicial independence, not even the five you named. For instance, the judges of the United States Supreme Court are nominated by the President and confirmed by the Senate. No matter how independent they may be afterwards, the two other branches of government have control about who becomes a judge to start with. As recent history shows, this can be a highly partisan process.
There is the habit to see advantages, disadvantages, and justifications of the own legal and political system quite clearly, while getting only distorted views of the rest of the world. That's a reputation many Americans have, fairly or unfairly.
- A snarky comment would be that you need a jury separate from the judge only if you don't quite trust the king's judges, but you didn't manage to get rid of them completely, either.
- Would you say that a city like Ferguson should be allowed to organize independent courts, or wouldn't the citizens benefit if there was a bit more supervision of the municipal court procedures?
- Is a freely elected judge in a U.S. state court who is coming up for reelection in a couple of months truly more independent than a career judge under the European system who will serve until retirement age?
So, to get to your question:
The European states developed as modern democracies because they had the rule of law, administered by a professional civil service and a professional judiciary. Such a judiciary comes in different forms and traditions.