The problem is that this is hard, very hard.
Take for example of only including the first half of the postcode, that's pretty anonymous, unless of course you have multiple postcodes (home and work, home and holiday home) at which point you will start getting unique or near unique combinations - particularly when you start adding year of birth in.
In isolation that data set may not be a problem, but combined with another one (land register maybe or just knowledge from facebook/friends) you can start to identify some classes of people.
With those people you may then be able to de-anonymize your health provider location (presumably it is a consistent mapping otherwise it is useless), at which point you can then start to identify more people.
Your main point is correct though, unless it has been successfully aggregated and combined much of data should just not be passed.
