because the rumor has it that "Edge sends full URLs of pages visited to Microsoft" and we haven't yet caught google in such embarrassing situation
Well, by default, Chrome sends whatever you're typing in the URL bar to Google for auto-completion. No need to catch Google in such situation, it's a feature of Chrome.
Edge does that bit too. Things typed in the URL bar are only a small subset of "pages visited", typically would not include session IDs etc, ...