Audit the codebase for UI blocking state system use cases.

## Feature Description

Conduct a comprehensive audit of Site Kit to identify where the UI blocking state system (introduced in #11297) should be applied, define the implementation approach, and coordinate the rollout through sub-issues.

While the initial implementation focused on the `UserMenu` and `DashboardSharingSettingsButton` components being disabled during the `isAutoCreatingCustomDimensionsForAudience` operation, there may be other components that trigger modals and other async operations that might display modals which should also use this centralised blocking system to prevent modal conflicts.

This encompassing issue will coordinate the audit process and manage sub-issues for migrating existing tightly coupled components, implementing the blocking system in new scenarios, and documenting cases where the system should not be used.

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## Acceptance criteria

- A comprehensive audit has been completed of all UI components that can trigger modals (buttons, menu items, etc.) and all async operations that might display modals.
- Sub-issues have been created for migrating any existing components with hardcoded checks for specific operation states to use the modal blocking state system.
- Sub-issues have been created for implementing the modal blocking state system in new scenarios where modal conflicts could potentially occur.
- Sub-issues have been created for refactoring any components that currently have tight coupling due to modal conflict prevention.
- All sub-issues have been completed and the UI blocking state system has been fully rolled out across Site Kit where appropriate.

## Implementation Brief

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### Test Coverage

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## QA Brief

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## Changelog entry

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