Yes, but their business is search advertising. You can get really good targeting with just 1st party data (what the current user has searched for).

That's completely different from display advertising. For most sites on the web there's not enough information about a user's interests on that site alone to get a lot of different bidders for the ad slot. Fewer bidders generally means the a list auction price, so less money for the publisher.

It's not just search advertising, that was in the early 2000s. Their business now is to profile you anywhere, anytime. That's why they bought Youtube, Android, Fitbit, and the reason for their every move.

Google does a hell of a lot more than search advertising. Why do you think they bought YouTube? They also power in-app ads, too.

It's expensive to collect, maintain and collate data yet they're making a profit on it. If they didn't want the data they wouldn't collect the data.

Regarding Google's business model; this problem gets fairly complicated in fact when it comes to the question that "what makes Google search alive?". Yeah, people go Google to search something and this loses its point without the open web ecosystem. Google's Display ads business certainly contributes to only a small fraction of its overall revenue but that's not the point of this business. Instead, its very existential reason is to remove paywalls from the web so Google search can remain useful.