In the new issue view you can now drag and drop subtasks in all issue detail view (board, backlog, full page, issue navigator). You can see the linked video in the attachments for how this works. Therefore, I am marking this issue as complete.
The new issue view has been rolled out to all Jira Software cloud customers with the ability for each user to individually toggle it off or on. You need to have the new issue view toggled on to have this functionality.
It should be noted however, that drag and drop for issues within an epic will not work. This is due to the fact that regular issue types - e.g. stories, bug, task - are tied into the global rank which can affect your backlog list. The subtask issue type(s) did not have this problem as they were not tied to the global rank. For example, if you moved the list of stories under an epic around, it would affect how those issues are represented in your backlog inadvertently and ways you may not have intended, causing confusion. We are designing the best experience for being able to drag and drop issues within an epic, while taking this into account. This will be shipped at some point in the future - no timeline yet unfortunately.
I have created a separate issue to track that functionality specifically. https://jira.atlassian.com/browse/JRACLOUD-72887
tpechacek (at) atlassian (dot) com
Product Manager, Jira Cloud