That's because the clipboard contains the "public.html" flavour -- Word is effectively translating the HTML from the page, downloading any images, and recreating the web page in the document. You can't write that flavour to disk, so you are left with the text only.
If you don't want the whole page then your best bet is probably to automate the "copy, paste into a new Word doc, save Word doc, close Word doc" that you are doing manually at the moment.