I can't say with certainty there wasn't some kind of "pharmacy benefits manager" behind the scenes, but everything about the transaction felt simple and like a standard claim. Point being, it's not obvious to me that "all systems" require an entity to handle pharmacy benefits in the way you seem to be saying.
A PBM is just the department of a health insurance company which negotiates with pharmacies and medicine suppliers. Pharmacies are required because society decided a qualified person should be double checking the chemicals that get prescribed to people by a doctor.
The health insurance company is an agent on behalf of an unknowledgeable and unable buyer that negotiates healthcare prices and (ideally) adjudicates the care itself to prevent waste or fraud. They would be better labeled managed care organizations (MCO), because people pay them to manage their healthcare in a sense, on top of providing insurance against expenses over the out of pocket maximum.
Technically, one or more of all the managed care organization's functions can be performed by the government as a single entity for everyone, and is in many countries. However, the US has decided to go with a very fractured approach, delineating large portions of the population into various tribes that receive various quantity and quality levels of healthcare that is adjudicated by various administrators. On top of this is 50 states with 50 regulatory bodies with 50 different rules around healthcare delivery.
Hence, there are a lot of systems and negotiations flying around and a lot of variance in delivery of healthcare.
