Reconcile the Application Builder lifetime model with Generic Host and #11415

## Objective

Research the current WinForms message-loop and `ApplicationContext` lifecycle, the Generic Host contracts, and #11415. Recommend one coherent ownership model for UI-thread startup, synchronization context, host start/stop, application exit, and failure propagation. Record alternatives and the smallest public contract needed by the other workstreams.

## Completion guidance

- Keep the implementation focused on the objective while allowing the assignee to improve the approach based on research.
- Document important decisions, rejected alternatives, dependencies, and follow-up work.
- Add or update tests and documentation appropriate to the work.
- Verify locally and in CI where the task changes code or generated artifacts.

Start by reading the current WinForms message-loop and ApplicationContext lifecycle, the Generic Host contracts, and issue #11415. Produce a recommendation for ownership of UI-thread startup, synchronization context, host start/stop, application exit, and failure propagation, including alternatives, decisions, dependencies, and follow-up work. Done means the smallest public contract is documented and appropriate tests and documentation are added.

Written by the indexing model from the issue text.