The menu is grayed out because there is nothing to save as it was just saved. I don't fully get why we would have such an option? It seems to be a situation that doesn't occur so often? If you don't what files to be saved automatically, just disable the option and save screenshots via shortcut CTRL+S

If we had a gallery of screenshots that we took, that would make more sense in my opinion.

Can you add a delete button to the tab? If it's not possible to add buttons to the tab, can you catch a keyboard modifier + a click on the close button? Eg, Alt + click on the close button = delete .
Or maybe if you can add arbitrary buttons to tabs, then one Close+Discard button and one Close+Save button.

We could add additional context menu entries to tabs, maybe with keyboard shortcuts for deleting screenshots that have been saved.

This is implemented now. You have an option under edit to delete the current image and in tab context menu items to delete others.

I also want this feature for quickly managing the screenshots. Although the delete option has been added, it might be still cumbersome. Just wondering whether a hotkey can be assigned to the deleting behavior, such as Delete.

I am using ksnip 1.9.1 on Ubuntu 20.04.

The Delete key is already used by the annotator to delete annotation items. But another combination could be done eventually, you can log a new issue and with a suggestion what combination to use, maybe CTRL+Delete