It does the most you can reasonably do: automates the native functionality. I wrote such a script myself for Tampermonkey but it was a tad janky.

Some suggestions for the dev:
1. Rather than popping open the menu, use the 'Ctrl + Shift + C' keybinding to bring up the word count dialogue.
2. If you can find a way to make the dialogue invisible, that would be great (I couldn't get it).
3. For whatever reason, doesn't work reliably if the document has heading text on the first line. Not every document is affected; I couldn't find the exact conditions to reproduce.