Refactor DiagnosticEngine to support multiple registered diagnostic handlers.

This fixes a problem with current save-restore pattern of diagnostics handlers, as there may be a thread race between when the previous handler is destroyed. For example, this occurs when using multiple ParallelDiagnosticHandlers asynchronously:

Handler A
Handler B | - LifeTime - |    Restore A here.
Handler C | --- LifeTime ---| Restore B after it has been destroyed.

The new design allows for multiple handlers to be registered in a stack like fashion. Handlers can return success() to signal that they have fully processed a diagnostic, or failure to propagate otherwise.

PiperOrigin-RevId: 270720625