Improve error handling for NeonAPI fixture

[tristan957]

Move error handling to the common request function and add a debug log.

[bayandin]

Looks nice, thanks!

[github-actions]

5426 tests run: 5199 passed, 0 failed, 227 skipped (full report)

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Flaky tests (4)

Postgres 17

• test_pg_regress[None]: debug-x86-64
• test_sharding_split_failures[failure2]: debug-x86-64
• test_scrubber_physical_gc_ancestors_split: debug-x86-64

Postgres 16

• test_timeline_archive[4]: release-arm64

Code coverage* (full report)

• functions: 31.8% (7890 of 24837 functions)
• lines: 49.4% (62452 of 126301 lines)

* collected from Rust tests only

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_{The comment gets automatically updated with the latest test results
9c2c10b at 2024-11-13T22:12:46.109Z :recycle:}