I agree that the Ufopaedia is an awkward place for this kind of info, but I can't come up with anything better.
Can you please explain how your idea would work? Where be the button at?
The button would be directly on the Geoscape (same as intercept, bases, graphs, options, funding, etc.) but there is no more space there
So at the beginning I would do only a hotkey, without a button.
Then a new GUI opens, which has a list of all items from all sources with their global count.
You can scroll through the list and mark/unmark some items (with a simple mouse click) as "watched".
On the bottom there would be OK button to close the GUI and also a [Show only watched] toggle button, which would filter the (very long) list and show only the items you have marked as watched.
The watched items would be saved in the save file, so that you don't have to select them every time of course.