You're right that people don't understand the way the separation of powers works, but the system badly needs to be revised. The world has totally transformed in the last 75 years and it's much smaller now. Political and legal feedback loops need to become much more responsive. One can see how it all worked when it would take weeks or months to get news of things like national election results, and when most people lived mostly self-sufficient agrarian lifestyles, but we don't live in that world anymore.
Our entire political system, including the functional ability for normal private individuals to seek and obtain useful legal relief, has ground to a halt over the last several decades. Technology has made the old methods increasingly bad fits for the modern era.
It'd be great if we could renovate the system before we fully convince the populace that the elements of good governance, like separation of powers and an independent judiciary, are necessarily linked to stunning and clear deprivations of fairness (of which binding arbitration is just one good example).