Depending on the size of the health system it may not be a team of multiple FTEs but they absolutely do expend significant resources on managing differences between commercial payers. They all have different rules about covered services, step therapy, prior authorization, hospital admission, etc. Sometimes those differ significantly even between health plans offered by a single carrier.

Many providers report that dealing with Medicare has clear rules they can follow while dealing with private payers causes a huge burden.

This isn't really true anymore (if it was ever true). Providers are spending a huge amount of time dealing with prior authorizations and appeals for private insurance.

I work in this area and you're right that Medicare can require a huge amount of paperwork from providers. And a hospital will have far more than 2 FTEs for this (it's called Revenue Cycle Management).