Your model doesn’t eliminate politics, it just entrenches the particular politics of the kind of people who go to T10 schools then get jobs as division heads at federal agencies.[1]

There is no such thing as “following the rules” in an apolitical way. Congress writes very broad laws, and the executive branch exercises a tremendous amount of discretion in enforcing and executing those laws. The founders understood that, and their solution to the problem was frequent elections, not the fiction of neutral, apolitical credentialed bureaucrats.

[1] A good example of this is the bank bailouts in the first Obama administration. Even though the voters were outraged at Wall Street, Obama followed the bailout strategy developed by Wall Street. He replaced Hank Paulson (Goldman Sachs) with Tim Geithner (NY Fed then private equity), but everyone underneath stayed the same and the bailout strategy stayed the same.

It's not a fiction. People do their jobs even if they don't like the current or past President. I'm sure you can pull out a long list of people who didn't, but unless you name everybody, it simply isn't a fiction. My claim isn't bureaucrats are always apolitical, it's that they mostly are. Showing that some aren't doesn't show that they mostly are.

Take the executive assistant to the American diplomat to, say, Sweden. They file paperwork and schedule appointments for the diplomat. Their role is operational. Logistics stuff. Coordinating what goes where. Setup a meeting between three very important busy people and juggle their calendars. Does that position really need to be someone we vote for? They do operations, not make policy.