[Cleanup] - Remove the `private_key()` method

[ss-es]

What is this task and why do we need to work on it?

We really should not be propagating private_key throughout the code the way we are. I'm pretty sure we don't do anything wrong with it (and the type system does guard against mishandling it somewhat), but it's generally bad practice.

What work will need to be done to complete this task?

No response

Are there any other details to include?

No response

What are the acceptance criteria to close this issue?

Remove the private_key() method. SystemContextHandle should provide a sign method that we can call as handle.sign(...) (or similar).

Branch work will be merged to (if not the default branch)

No response