That is a minor issue. Most public healthcare systems are not british-style NHS, and healthcare providers (doctors, hospitals) are independent of healthcare pay

That is a minor issue. Most public healthcare systems are not british-style NHS, and healthcare providers (doctors, hospitals) are independent of healthcare payers (public health insurance orgs) and are often private. The main advantage of public heathcare is that being single-payer it can use its monopolistic buyer power to push prices down. It leads to lower overall costs, but also to underinvestments and lower wages in healthcare.