Link to Issue or Description of Change

• Closes: Loosing SSE session from the browser shouldn't break the agent run #4029

Problem:
Disconnecting a /run_sse client cancelled the active agent run. Long-running
tools stopped before completion and left incomplete events in the session.

Solution:
Run the agent lifecycle in a server-managed task and relay events to the SSE client through a bounded queue.
A disconnected client stops receiving events, while the agent run continues and persists its remaining events.
Active tasks are cancelled during server shutdown.

Testing Plan

Unit Tests:

• I have added or updated unit tests for my change.
• All unit tests pass locally.

Ran:

uv run pytest tests/unittests/cli/test_fast_api.py
127 passed, 5 skipped, 1 xfailed

The regression test closes the SSE response after its first event and verifies
that the underlying agent run finishes without cancellation.

Manual End-to-End (E2E) Tests:

1. Started adk web with a temporary agent containing a deterministic model
   and a 20-second asynchronous tool.
2. Created a session and started /run_sse.
3. Disconnected the client after the tool started but before it completed.
4. Waited for the tool to finish and fetched the session.
5. Sent a follow-up request on the same session.

Observed:

• The tool completed and was not cancelled.
• The session contained the user message, function call, function response,
  and final model response.
• The follow-up request completed successfully.

Checklist

• I have read the CONTRIBUTING.md document.
• I have performed a self-review of my own code.
• I have commented my code, particularly in hard-to-understand areas.
• I have added tests that prove my fix is effective or that my feature works.
• New and existing unit tests pass locally with my changes.
• I have manually tested my changes end-to-end.
• Any dependent changes have been merged and published in downstream modules.