Sounds like they're half as useful to the business rather than their jobs are twice as hard

Effort and business utility are not necessarily linear. For argument's sake let's assume they are. A manager affects the productivity of all of his/her reports. Let's say a manager has 10 reports. If the manager stops working twice as hard, maybe all of the manager's reports would become 1/2 as productive. That means the manager's 2x work provides the same business utility as 5 ICs.

This assumes that manager productivity is positively correlated with their reports' productivity.

It's hard for me to think about how else to measure manager productivity. Managers don't usually submit code themselves.

Lots of people in a business aren't submitting code, but that's not really my point. Depending on a manager's style they can increase synchronization overhead between team members and different teams to the point that everyone's productivity is reduced while the manager has never been busier or productive on paper.

A manager's productivity IS the team's productivity, and should be measured in results.

A "productive" manager that slows down the team isn't an effective manager at all, no matter how hard he or she is working.

The manager might be able to scapegoat a poor team member or two when poor results become evident, but sooner or later, he or she will have to pay the piper.

I used to think that before I became a manager.