Without their search engine, there is no advertising business.
They do have significant advertising businesses beyond search (AdSense, gmail, etc.).
My understanding is that if search and ads were separate businesses and ads paid search for search result ads at market rates then search would not be profitable (similarly for e.g. Gmail). (In reality if search were an independent company then it might be able to be profitable by selling information to ads in addition to direct advertising revenue - OTOH the legal barriers to selling information to a third party might make that impractical, whereas conveniently they're not an issue while Google remains a single company)
[1] Page 24, https://investor.google.com/pdf/20141231_google_10K.pdf
While I don't agree with the parent you're implying that search is the ONLY Google property that displays ads but that's not true. A decent percentage of that is coming from gmail. I'm not sure how much however.
What? I have never heard that AdWords runs at a loss and find it incredibly dubious.
I'll just say that's a convoluted definition of a loss-leader. If a revenue line requires an expense, that expense is not a loss-leader, it's just a cost. That's like saying that having a website is a loss-leader for Amazon, since they really make their money on selling products.