Backport of release log gate if disable-gated-logs flag is set into release/1.13.x

[hc-github-team-secure-vault-core]

Backport

This PR is auto-generated from #24280 to be assessed for backporting due to the inclusion of the label backport/1.13.x.

:rotating_light:

Warning automatic cherry-pick of commits failed. If the first commit failed, you will see a blank no-op commit below. If at least one commit succeeded, you will see the cherry-picked commits up to, not including, the commit where the merge conflict occurred.

The person who merged in the original PR is: @hghaf099 This person should manually cherry-pick the original PR into a new backport PR, and close this one when the manual backport PR is merged in.

merge conflict error: POST https://api.github.com/repos/hashicorp/vault/merges: 409 Merge conflict []

The below text is copied from the body of the original PR.

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Vault wouldn’t output any logs at all on startup right now as the logs are gated to be flushed until after lot of initialization is done. This PR introduces a hidden flag which is mostly used for troubleshooting that allows to flush the logs as they become available. Addresses https://hashicorp.atlassian.net/browse/VAULT-20508

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Overview of commits

• 57b6b742838f117460d538359c3f6382d1e7dab5

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CI Results: All Go tests succeeded! :white_check_mark: