[Healthcheck AI][Task 2] Add the public Healthcheck assistant extension and drawer layout

Parent: #4186
Related follow-up: #4187

Part 2 of 3 for the Healthcheck → AI assistant flow.

Scope

Expose the Healthcheck integration from the public UI package:

add an optional no-op HealthcheckAssistantAction component-registry slot;
pass a typed target and the already loaded Healthcheck snapshot;
place Diagnostics next to status/refresh only for a successful snapshot with issues;
support the per-issue Fix action required by Implement AI assistant buttons for Healthcheck issues #4187 without toggling the issue row;
preserve semantic cluster/database scope and the exact request used by the drawer;
add a controlled drawer rightInset so Healthcheck and AI panels can stay side by side and remain resizable.

The default slot must remain a no-op so importing the package alone does not change the UI.

Cluster and database targets and snapshots are covered by tests.
Loading, error, GOOD/no-issues, and issues states preserve the intended visibility.
Diagnostics and Fix placement and click propagation are covered.
Inset, resize, cleanup, percentage width, zero-width, and persisted-width behavior are covered.
Lint, stylelint, typecheck, tests, and package build pass.