Add default autosave restore UX

[adamziel]

What it does

Makes browser storage autosave the default Playground experience when available.

Users get a restore nudge for recent autosaves, the save status reflects
autosaving/finalized states, and the Save modal presents storage choices
consistently around the current persistence state.

Rationale

The UI should not make users decide between browser storage and local directory
storage unpredictably. This PR keeps the default autosave flow separate from the
overlay redesign so the main user-facing state transitions are easier to review.

Implementation

Updates the selected-site guard, save status indicator, Save modal, Site Manager
details, and the query API documentation/tests for default autosave behavior.

This is stacked on feature/autosave-site-lifecycle-api because it consumes the
autosave persistence model and lifecycle actions added there.

Testing instructions

npm exec nx run playground-website:typecheck
npm exec nx test playground-website --testFile=setup-url.spec.ts