I'm organizing my board using stories and subtasks within those stories. This way is very helpfull to have an organized view of what is still pending for a story to be completed or done within Active sprint view. But it will be great to have something similar within backlog view so it could be easier to know which members of the team have subtasks assigned for that story in a single sight.
It looks like the default scrum backlog doesn't know subtasks, but the kanban backlog does. Which project type are you using?
You might be able to force the backlog view to show tasks by editing the filter query. (https://confluence.atlassian.com/jirasoftwarecloud/configuring-filters-764477988.html)
In the past, my team has used something like this
project = projectKey and issuetype in (Story, Bug, Task, Sub-task)
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