EXPERIMENTAL: do not close binlog connection on commit error

[shlomi-noach]

Description

Looking to see if/which tests fail if we do not close the connection on commit error. I suspect nothing will fail in the tests, because this particular error is so rare. Let's see what happens and continue the investigation from there on.

Related Issue(s)

https://github.com/vitessio/vitess/issues/17248

Checklist

• [ ] "Backport to:" labels have been added if this change should be back-ported to release branches
• [ ] If this change is to be back-ported to previous releases, a justification is included in the PR description
• [ ] Tests were added or are not required
• [ ] Did the new or modified tests pass consistently locally and on CI?
• [ ] Documentation was added or is not required

Deployment Notes

[vitess-bot[bot]]

Review Checklist

Hello reviewers! :wave: Please follow this checklist when reviewing this Pull Request.

General

• [ ] Ensure that the Pull Request has a descriptive title.
• [ ] Ensure there is a link to an issue (except for internal cleanup and flaky test fixes), new features should have an RFC that documents use cases and test cases.

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• [ ] There should be a comment at the top of each new or modified test to explain what the test does.

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• [ ] RPC changes should be compatible with vitess-operator
• [ ] If a flag is removed, then it should also be removed from vitess-operator and arewefastyet, if used there.
• [ ] vtctl command output order should be stable and awk-able.

[codecov[bot]]

Codecov Report

All modified and coverable lines are covered by tests :white_check_mark:

Project coverage is 67.40%. Comparing base (3499a77) to head (88b8b61). Report is 3 commits behind head on main.

Additional details and impacted files

```diff @@ Coverage Diff @@ ## main #17253 +/- ## ======================================= Coverage 67.39% 67.40% ======================================= Files 1570 1570 Lines 252903 252933 +30 ======================================= + Hits 170451 170489 +38 + Misses 82452 82444 -8 ```

:umbrella: View full report in Codecov by Sentry.
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[shlomi-noach]

As expected, all checks pass.