Resolve "Pipeline view stages showing incorrectly"

What does this MR do?

1. When the dropdown is clicked, emits an event to the main component.
2. If a request is in progress, it cancels it
3. Stop polling
4. Fetches the table
5. Restarts polling

Does this MR meet the acceptance criteria?

• Changelog entry added, if necessary
• Documentation created/updated
• API support added
• Tests added for this feature/bug
• Review
  • Has been reviewed by UX
  • Has been reviewed by Frontend
  • Has been reviewed by Backend
  • Has been reviewed by Database
• Conform by the merge request performance guides
• Conform by the style guides
• Squashed related commits together
• Internationalization required/considered
• End-to-end tests pass (package-and-qa manual pipeline job)

What are the relevant issue numbers?

Closes #40487 (closed)