One factor that isn’t being explicitly mentioned is that some countries like France and the United Kingdom have a tradition of putting non-political, expert government bureaucrats in positions of authority to check the power of elected and appointed officials.
We do have a civil service in the United States, But it’s much more at the mercy of elected and appointed officials and can’t overrule them.
So that’s one reason why Americans can’t conceive of a system in which prosecutors and judges arise through non-political bureaucracies and and aren’t beholden to popularity or other political winds.
Put that together with the strong cultural streak that distrusts expertise, intellect, and any other objective, non-popularity-based standard for putting someone in charge.
AIUI, the concept of non-partisan professional public service, in the UK at least, dates more or less from the mid 19th century, becoming more powerful as a result of the First World War. Which is not to say that ‘jobs for the boys’ disappeared overnight, or that the public service culture wasn’t beset with assorted prejudices and assumptions that have in turn needed to be educated out, which is probably a never-ending job.
But it does mean that interference by politicians in individual cases, or in established processes, can get them into a lot of trouble. That basic concept is pretty widely established in the public consciousness.