Advertising is your core business when almost 100% of your revenue comes from that. I wouldn't call what turned it from a dorm room project into a behemoth a "tack on", that's dismissing Eric Schmidt's contribution. What you do to achieve this goal doesn't matter as much. Amazon core business is e-commerce but they rent servers and sell portable devices.
> Ad relevancy is really just a natural extension of their Search algorithm.
You can argue that advertising is a natural extension of everything they put their hands on, from search to social networks to maps to glasses with HUDs. Everything they develop lately has advertising potential, and are shutting down everything that doesn't.
AdWords predates Eric Schmidt joining the Google board, let alone becoming CEO.
The rest of your argument is just silly reductionist wankery. Whatever the "core" of their business really is, they do have people working on those things. If you're going to make an incentives argument, make it; arguing where to put the "core" label provides as little insight as repeating "you're the product" ad nauseam.
The GP is exactly right. Yes, you do have incentives to maximize for advertising revenue. But if you want to maximize revenue over any length of time beyond the next quarter, you neglect things like having a product that's actually useful, usability, intrusiveness of ads, trust in your ability to keep information private from advertisers and the NSA, etc etc at your extreme peril. You can have all the advertisers in the world and it does you no good if everyone stops showing up to look at their ads.
Also, I was just pointing out the dichotomy between the core competency and core business (what you do well vs. your source of revenue).