I just tried this from the NYTimes site (copying some text from an article in Chrome and pasting into Word, then going back and copying the URL, all using the mouse and context menus not the keyboard shortcuts) and it went fine. This is using Word 2007 and while there's no end to Microsoft's work of adding questionable new "features" in subsequent versions, I'd be surprised if that mattered.
In fact this sounds like more of an issue with Chrome, since presumably if you pasted into anything else (Notepad, say) you'd still be getting the text rather than the URL -- if indeed the URL copy didn't "take." And if it didn't "take" then that would be a Chrome (or add-in?) issue, no? A long time ago I had an issue with Chrome where it wouldn't reliably replace one URL in the clipboard with another, resulting in me pasting old links rather than new ones, but that went away with an update years ago and I haven't seen anything like that since.
BTW, you might want to try one of the various "scrapbook
" extensions for Chrome (none of which I have tried), though if saving things in Word works for you I'm not going to argue (especially if you're planning to use and cite the material in work you're writing in Word anyway).