This hierarchy below is an example of a higher level being added to the default hierarchy:

I would like to be able to add a level below stories and remove sub-tasks. The reason for this is that stories can be used to represent requirements, but may or may not be able to be completed in a sprint. I would like to test these requirements via manual and automated testing at some point when they are finished, and complete tasks in sprints.

JIRA doesn't deal we well with having items span multiple sprints, so I would like the tasks to be the work items completed sprint to sprint. This use case assumes a story isn't a product increment that can be finished in a sprint, but a goal or deliverable that will be worked through over time. For example, a slow infrastructure build, that isn't following agile manifesto, but is benefiting from some agile features.