In Germany, we have a health insurance card. Let's say your insurer is Techniker Krankenkasse. They provide you a card with your photo and an NFC chip. You show this card in any doctor/hospital you visit, and your expenses are all paid. Today, as a new feature, your prescriptions are also stored to this card. You show the card in the pharmacy, get your medicine and the costs are all paid by the insurance company (minus the co-pay, 10 euros, which you pay by yourself).

Edit: To be clear, we don't have public hospitals or doctors. They're all private. But the insurance can be public, and the doctors and hospitals accept your public insurance and you don't need to pay for them.