The examples they list make trivial errors. The health data obviously doesn't need to contain any bits of someone's SSN, phone number, or their exact address.

Those trivial errors weren't made by them.

In general anonymisation is hard. Hard enough that there's an interest of lawmakers to regulate it which makes it hard to regulate.

At the moment it seems like Google, Microsoft and Apple all build their health data storages, and then those companies put their own machine learning people on the problem.