Fix log output in post-start scripts following backgrounded process

MR: Pending

Description

This is a followup to this discussion: !194906 (comment 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.

Acceptance criteria

• Outputs following backgrounded process show are logged correctly to poststart-stdout.log