An oncology department I'm familiar with has an entire "nurse navigator" whose whole job is to submit "prior approval" requests to "insurance" companies justifying why patients need a specific treatment, plus the nurses employed by the "insurance" companies reading those requests. I believe it's similar for any moderately expensive specialty. A common career path is care nurse -> burnout -> administration. Most of the administration is made up of people who could be providing healthcare.
And no, hospitals' most valuable resource are their billing computers. I think when it comes to providing actual healthcare hospitals are very stupid. You cannot partition any knowledge worker's attention into 10 minute blocks and expect them to achieve anything useful, yet that is what their entire system is designed around. The hospital doesn't have unilateral say of course (an "insurance" company won't pay one doctor the "price" of two if they spend twice as long with a patient), but they're still content optimizing within that status quo outcome - completely scatterbrained care.
And it's not like individual doctors are well rested or happy when you talk to them. The system clearly takes their toll on them (eg disappearing for 5 minutes to go retrieve test results that didn't show up before your appointment). In fact I'd say the vast majority of human talent in the medical system ends up completely wasted.