proxy: clean up code and lints for 1.81 and 1.82

[cloneable]

Problem

Summary of changes

Checklist before requesting a review

• I have performed a self-review of my code.
• If it is a core feature, I have added thorough tests.
• Do we need to implement analytics? if so did you add the relevant metrics to the dashboard?
• If this PR requires public announcement, mark it with /release-notes label and add several sentences in this section.

Checklist before merging

• Do not forget to reformat commit message to not include the above checklist

[stradig]

LGTM

[github-actions]

3836 tests run: 3722 passed, 0 failed, 114 skipped (full report)

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Flaky tests (3)

Postgres 16

• test_ondemand_wal_download_in_replication_slot_funcs: release-x86-64
• test_subscriber_restart: debug-x86-64

Postgres 15

• test_subscriber_restart: release-x86-64

Code coverage* (full report)

• functions: 31.9% (7346 of 23027 functions)
• lines: 50.0% (59482 of 118895 lines)

* collected from Rust tests only

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_{The comment gets automatically updated with the latest test results
5a2dc46 at 2024-09-06T15:15:19.679Z :recycle:}