Well, Shanahan is acting and will be confirmed when the senate gets around to it... that's kind of the point though. DOD is still running and will continue to run, exactly as it has, with only minor political differences from the top. They have a lot of theoretical power, but little ability to actually change day to day operations of anything.
The purpose of confirmed political appointees is to create a layer of empowered leaders who can do more than simply steer the ship in a straight line, or react slavishly to orders from above. The confirmation process serves two purposes: (1) it ensures that relatively independent thinkers with high political capital are in those spots, and they see their allegiance to the entire system and not just one man, (2) it provides a safety valve (via resignation) when the confirmed appointee does not agree with orders from above. The danger of the DoD is that it’s an agency that can give the appearance of running itself when there’s no crisis, but may need expert leadership when there is one. Since the entire purpose of the DoD is to manage crises, lack of high-level leadership is a serious concern. Ditto the FAA.
As I posted in response to your other comment, you're ignoring the fact that the undersecretaries who backfill those positions in the absence of the primary are also appointed and confirmed, and the career personnel have less incentive to bow to political pressure.
DOD has far more "expert leadership" in the form of the FO/GO community than the other gov agencies as well -- the secretaries exist to implement policy, not run day to day operations.
Oh, sure. I just enjoyed the combination of the example being a bit off, because it sounded like a hypothetical, but also a perfect real-world demonstration of the idea.