Fix log output in post-start scripts following backgrounded process
MR: Pending
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## Description
This is a followup to this discussion: https://gitlab.com/gitlab-org/gitlab/-/merge_requests/194906#note_2582822374
Some output lines don't show up in the `poststart-stdout.log` when there are backgrounded processes in the scripts. Figure out why this is so and fix it.
Example:
In `internal_poststart_command_start_vscode.sh`
`echo "$(date -Iseconds): Finished starting GitLab Fork of VS Code server in background"`
This does not show up in the logs.
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## Acceptance criteria
- [ ] Outputs following backgrounded process show are logged correctly to `poststart-stdout.log`
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issue