"At issue is the system's default privacy setting, which is that any health professional treating an individual can access their whole health history."
Maybe that's what's at issue right now, but it's not the real issue. The real issue is that the government does not value the privacy of the citizens and it does not want to give them any control over their own information.
That's why this trial was changed from opt-in to opt-out. If it's opt-in then people are more likely to actually figure out what it's all about and only actually sign-up if they believe it to be in their best interests - weighing the pros and cons.
If you go ahead and sign everyone up without their express permission then you don't have to worry about the cons - you just spend your time spruiking the alleged pros.
They focus only on getting people on and getting the positive results so they can trumpet them and pat themselves on the back. They spend no time really caring about the negatives so it's inevitable that problems like this are almost built in to the system.
Opt-in essential means: if you're not sure, don't do it. If you don't understand the pros and cons and what it means for you, then it's better if you wait until you do. Sure, many people will simply ignore it but that's why you do a campaign for it and make it super-easy to opt-in. Show them the benefits and treat any concerns about the negatives honestly.
When it comes to measures that reduce privacy, it should always be opt-in.