I think the govt has a point here. The public dealt with the health care providers and entrusted them with their data. It was the providers who chose to offload work to a third party. It should be those providers who are directly responsible to their customers. I regard that as an important point of principle*.
The providers certainly have a complain against their agent who let them down and may well be entitled to demand the agent do the work on their behalf and do it for free as it was their failure.
* It may be of less significance here but its disregard in the way transatlantic GDPR responsibilities are hidden is a real problem.
If an EU data subject is failed by the US service provider for the EU company with whom the subject is dealing they're expected to take it up in a US court with the service provider which is going to raise a substantial barrier compared with taking a court action in the jurisdiction, probably their own, in which the original transaction was made.