mandating a high base pay in the sector, and subsidizing it.

This is simply not a realistic option.

The current German health expenditure is €441 billion. (€8320 per working person)

Any such subsidy has to come from somewhere, and the amount of money involved means you can't just increase taxation a negligible amount.

When it's projected the healthcare workforce (in Germany: 6 million people) needs to DOUBLE in the next 20 years, you can't just get that done with some cheap incentives and "healthcare is cool!" posters.

The demographics of an aging population simply mean that if you want more healthcare workers, you need more immigration to get them.

If you don't want to increase immigration (as the german elderly population seems to vote), you have to cut healthcare availability. One or the other.