Oh sorry, my bad for not answering your questions!
If I understand you correctly you want to have a quick access to your saved search templates that you can share with others and run to the open Editor tab. Am I right?
Well, that would indeed be awesome, but at the least it would be great if I could rerun a saved search template myself. We can start with that
Do you want to have the way to run these search templates to the several Editor tabs?
I like your initiative. That was not initially not what I needed, but I am already realizing I could really use that as well!