Huh? You're the one adding the extra layer, by having the check sent to the recipient before they pay the provider. The number of layers required to cut a check to a hospital are the same as the number of layers required to cut a check to the recipient.
In Michigan the Medical Services Administration spends $43 million per year (mostly Federal) to run a $6 billion Medicaid program. That's an administrative cost of less than 1%, far better than that achieved by most private insurers (to be fair, that's because Medicaid is a much larger payer and there are economies of scale). The same is true in other states. Administration isn't the problem.