While on the Edit Person screen you can use Options>Put FamilySearch ID on clipboard" and then use a browser bookmark to open to FamilySearch's Find tab - https://www.familyse...rg/tree/find/id and paste the FSID there to search. It's a lot faster than waiting for FamilySearch to load.
For the Ancestry Tree a shortcut would be to make a custom search provider in WebSearch. Go to the Ancestry tree online and click on Tree Search and then the "List of all people". Enter John Doe in the search box and use the URL that is returned on the search for the WebSearch Search Results URL. Use the external browser option in WebSearch if you want it to open in the browser.
While this works, it still takes multiple clicks or keystrokes. Providing one click shortcuts from the main and edit person windows to launch the appropriate web page still seems like a worthwhile enhancement to me.