Presumably even in a single-payer system, you still need just as good of an accounting of costs and care provided. The data needs are all still there, just fewer parties are transacting.
In the USA, people can and do die because there's a financial barrier to accessing medicine. Not so in countries that care about people.
If the computers didn’t work, it would be incredibly painful in Germany because most pharmacies don’t regularly stock most prescription drugs. They’re often ordered from central warehouses on demand - i.e. if you walk in with a prescription for Xanax in the morning, you’ll typically need to come back in the afternoon to pick it up.
That ordering process is entirely digital. If you suddenly switched to paper forms, phone calls, and faxes, it’d be a nightmare.
An even more fun quirk is that because different pharmacies seem to use different distributors, you'll sometimes have to hop around until you find one whose distributor has whatever your medication is in stock. I once had to go to four different pharmacies in Mitte before I found one that could order Zolpidem.