ref(pydantic-ai): Use hooks.on.run()

[alexander-alderman-webb]

Description

Use hooks.on.run() to replace monkey patches.

Note that the gen_ai.response.streaming attribute is no longer set. Hooks do not readily expose whether the response is streamed.

Also add some more precise types.

Issues

Reminders

• Please add tests to validate your changes, and lint your code using uv run ruff.
• Add GH Issue ID & Linear ID (if applicable)
• PR title should use conventional commit style (feat:, fix:, ref:, meta:)
• For external contributors: CONTRIBUTING.md, Sentry SDK development docs, Discord community

[github-actions]

Codecov Results 📊

✅ 69119 passed | ⏭️ 3315 skipped | Total: 72434 | Pass Rate: 95.42% | Execution Time: 216m 38s

📊 Comparison with Base Branch

Metric	Change
Total Tests	📉 -66532
Passed Tests	📉 -62716
Failed Tests	—
Skipped Tests	📉 -3816

All tests are passing successfully.

✅ Patch coverage is 100.00%. Project has 2285 uncovered lines.
❌ Project coverage is 89.77%. Comparing base (bb891ea) to head (1d5726d).

Coverage diff

@@            Coverage Diff             @@
##        master       #PR       +/-##
==========================================
- Coverage    90.17%    89.77%     -0.4%
==========================================
  Files          193       181       -12
  Lines        25790     22342     -3448
  Branches      9532      8120     -1412
==========================================
+ Hits         23256     20057     -3199
- Misses        2534      2285      -249
- Partials      1434      1317      -117

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