For the sake of anyone reading this thread who may not be familiar with Properties Service.
Under the category of Properties Service is:
Document Properties - documents are Sheets, Docs, Forms and Slides.
User Properties - restricted to the user
Script Properties - restricted to the Apps Script Project file
This discussion is specific to Script Properties.
I guess if someone wanted to create an HTML UI and open source it,
you could add it to your project, and then run it as a Web App.
That would give you a UI for managing Script Properties.
Of course, if the Apps Script project was already deployed as a Web App,
then you'd need to modify something to get the correct HTML to load.
So, I can't think of a great solution to this issue if the legacy editor ever gets deprecated,
or if they remove the Script Properties tab from the legacy editor.
I'm guessing that Google might add a Script Properties management UI that you open directly
from a list of your Apps Script files, rather than through a code editor toolbar item,
since the Script Properties are directly tied to the file itself.
Maybe that should be a feature request if Google hasn't already thought of it?