A lot of the medical issues in Cuba aren't related to the healthcare system but rather to trade embargos. It's a small miracle they do as well as they do given the constraints of being an island nation.
The reason for their success on a shoestring budget is administrative competence. They have a large number of clinics (rather than big hospital complexes) and education in medicine starts at those clinics. Future doctors work and are educated in medicine moving up into specialties. It's a little like making everyone that wants to practice medicine start as a orderly in a family medicine clinic.
IMO, this is superior to the US system of requiring several years of schooling before ever interacting with patients. And if you know an old nurse, you'll know they often do know a lot more than new doctors.