WIP: Replace jQuery old-style ajax calls with Promise interface

What does this MR do?

Replace jQuery old-style ajax calls with Promise interface (Deferred objects).

Are there points in the code the reviewer needs to double check?

No.

Why was this MR needed?

Old-style jQuery Ajax calls should be removed as jQuery will not be removed in the near future.

Does this MR meet the acceptance criteria?

• Changelog entry added
• Documentation created/updated
• API support added
• Tests
  • Added for this feature/bug
  • All builds are passing
• Conform by the merge request performance guides
• Conform by the style guides
• Branch has no merge conflicts with master (if it does - rebase it please)
• Squashed related commits together

What are the relevant issue numbers?

Closes #29105 (closed)