Convert remaining reads and writes to Query and retire the auth store

[ExtraToast]

What to build

Server state moves into the query cache and the Pinia store that held it is deleted.

This application has one store, which is what makes it the right pilot: the conversion is small enough to review carefully and to serve as the reference for the two larger applications.

Acceptance criteria

• Reads are served from the query cache and refetch according to the shared policy
• Writes invalidate the keys they affect, and the affected screens update
• The store no longer exists
• Coverage thresholds still pass after the store's tests are removed with it

Blocked by

• Move every auth-ui operation onto the globally configured client