No mention in that document how to:

  1. Search link text only.  
  2. Search link URL only.  
  3. Search headings only.  
  4. Search under the current heading only.

I don't know about the first two because those aren't a use case for me, but for 3, you want helm-org-rifle-agenda-files, and for 4, I don't know about something org-specific, but you can use emacs' search commands after narrowing the focus to the current subtree (by default bound to C-x n s, I use this so often that I have it bound to H-f, and I revert focus with s-f, so that I am not distracted while working on an entry).

The first two are interesting use cases though, I'll reply again if I find something that works.

For 3 and 4, this seems to make it possible: https://kitchingroup.cheme.cmu.edu/blog/2017/01/03/Find-stuf...

I, for one, will now go down this rabbit hole ...

So can you mention at least one way to simply search?