This is very wrong. Google cares deeply if Chrome wins. You are being way too narrow in your focus.

Advertising is not their product, it's their revenue stream. Their product is _everything_ that sits between a user and monetizing an ad. Just look at what they have done from an engineering, offensive and defensive position here. It's all about that ethereal space between users and monetization. That's the battlefield, and they want all of it.

They need FF because when Chrome dominates, FF needs to be a viable competitor. With Bing, it would be worse off.