The CEO has an incentive to find cheaper ways to deliver a service because they pocket some of the savings. A government employee gains nothing if they find a cheaper way of delivering a service — their compensation is fixed.
What exactly is there to improve in issuing health coverage? It’s just pooling money that people put in and handing out payments here and there. It’s not really something you can improve. Ford can make better cars. Farmers can grow healthier food. Money is just money and doesn’t get better. You can manipulate rates, but that is an art that isn’t going to improve. After a certain point, health care is just rent-seeking, and the $100 million is administrative waste, the very kind that you claim to hate. You are actively praising the problem that you have identified.