Migrate to FastAPI lazy router inclusion

[shaealh]

Addresses item 2 of #69616.

Summary

• Raise the Cadwyn minimum to 7.1.0 and FastAPI minimum to 0.137.2.
• Replace mutation of a shared Execution API router with fresh, build-time router assembly.
• Use FastAPI's iter_route_contexts() for authentication and token-scope route inspection.
• Preserve custom token-scope metadata from the original routes.
• Add a negative regression test proving that an accidentally unprotected route is detected.
• Refresh uv.lock to Cadwyn 7.1.0 and FastAPI 0.139.0.

Testing

• Full Task Execution API suite: 524 passed.
• Focused FastAPI migration suite: 36 passed.
• Airflow core mypy: passed, 1,202 files checked.
• Ruff formatting and lint: passed.
• API server startup: verified.
• Dag processor startup: verified.
• git diff --check: passed.

Gen-AI disclosure

• Yes, generative AI tooling was used. -- OpenAI Codex

[github-actions]

uv.lock on main just moved via #70363 ("[main] Upgrade important CI environment"), commit c3e2439 and this PR currently conflicts.

Quickest fix:

git fetch upstream main && git rebase upstream/main
rm uv.lock && uv lock
git add uv.lock && git rebase --continue
git push --force-with-lease

Automated nudge — ignore if you're not ready to rebase. This comment is updated in place on future uv.lock bumps.