ref(mcp): Replace instance patches with middleware

[alexander-alderman-webb]

Description

Add a middleware instead of wrapping handlers on registration.
All requests go through the middleware, whereas there are multiple ways to register a handler.

Issues

Closes #6986

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 📊

✅ 98904 passed | ⏭️ 6479 skipped | Total: 105383 | Pass Rate: 93.85% | Execution Time: 358m 55s

📊 Comparison with Base Branch

Metric	Change
Total Tests	—
Passed Tests	📈 +1
Failed Tests	📉 -1
Skipped Tests	—

➖ Removed Tests (1)

View removed tests

• test_continuous_profiler_auto_start_and_stop_sampled_span_streaming[experiment-thread]
  • File: tests.profiler.test_continuous_profiler

All tests are passing successfully.

✅ Patch coverage is 87.14%. Project has 2504 uncovered lines.
✅ Project coverage is 89.94%. Comparing base (base) to head (head).

Files with missing lines (1)

File	Patch %	Lines
sentry_sdk/integrations/mcp.py	87.14%	⚠️ 18 Missing and 20 partials

Coverage diff

@@            Coverage Diff             @@
##          main       #PR       +/-##
==========================================
+ Coverage    89.92%    89.94%    +0.02%
==========================================
  Files          193       193         —
  Lines        24859     24896       +37
  Branches      8962      8978       +16
==========================================
+ Hits         22354     22392       +38
- Misses        2505      2504        -1
- Partials      1414      1432       +18

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