To be honest, in general it does improve battery life if you are faster. Battery life is a race to CPU sleep mode, and if you finish the job faster, the more energy you can save.

Only for things that are "completable", not for things that run in the background persistently

Not much runs persistently, more like periodically. And the amount of time it has to be running still matter — processors when turner on leak a bunch of energy no matter what.

I’m skeptical about this. Why would a faster processor in it of itself be more energy efficient at executing instructions?

A faster processor reduces execution time allowing the CPU to return to sleep status faster. Total energy used is power x time. While increasing operating frequency increases power quadratically due to P = CV^2, it is usually a net energy savings to let the frequency boost as high as a mass produced chip can go to minimize time spent in the highest power state. This assumes a finite bursty workload.