These tokens have to be worth something in order for the security to functionally exist. You can't separate the monetary side from the security, because the monetary side incentives the security. And security is the only thing blockchain adds.
Not wanting it to be public should be obvious. Immutable though, what if I need to change my name/gender to match reality?
It's not enough to just update the value, because the old value still exists on the blockchain. That's just another method to find my deadname and use it for harassment.
But the points you raise, are exactly the issues I've been thinking could be solved with web3. I am imagining using it to give control to the patient of who has read access(to what and when), who can add data, etc.
I.e. give full transparency and control to the patient. Instead of the current situation where a patients data is on different systems, you don't know what it actually says, besides what a doctor tells you.