It's the add-on I've been looking for!
1. It allows to keep tabs in a structured way: it creates a dedicated folder for each separate saving.
2. It allows customization of folder's location.
3. It gives an option to close the saved tabs or to keep them
4. You can give a custom name to the folder.
5. It saves the tabs which are not hidden (most beneficial for those who use tabs grouping add-ons like Panorama Tab Groups).

The developer is most responsive. I asked via github for a feature. He wrote me a couple of letters that same day. The next day a new version of the add-on was posted, and he even sent me a demo video.

Dear Igorlogius, you are a wonderful developer! Wish you good luck!

Does what it says, no problemo. Some additional options would be nice (Changing the default location, folder-naming convention, etc), but out of the box it functions like a GUI free Tab Stash, which I can get behind.