I think leadership at Google back then truly did like it. And they continued to "mostly" like it up to about 7-10 years ago. During the time I was there (10 years from 2012 until end of 2021) I feel like I saw the transition happen, slowly. It got really contradictory and messy.
In fact, Google Plus (2011-2013ish) was justified (in part) as an attempt to "rescue" the web from the enclosing grasp of the walled garden that is Facebook -- which couldn't be indexed, searched, and presented openly. (Naturally this was hypocritical [in addition to being a failure] but that was one of the motivations given for the project)
Overall I think all of this had a lot to do not only with Google's positioning as an advertising company, but with the rise of mobile computing and mobile apps. The writing was on the wall when mobile use started to surpass desktop browser use. "The web" as we traditionally knew it ceased to be the primary interaction mechanism people were having with the Internet, and Google (and others) had to pivot quickly or they would lose the advantage they had.
Google from 2000-2015 or so seemed to innovate on "the web"; from search to gmail to news to reader (and even arguably to g+) it was about pushing the limits of the tech and excelling at making WWW products. Then it became about making... nice apps. I guess?
Story time: I worked on the Google (then "Nest") Home Hub. The original UI for it was written in HTML/CSS/JS and it was very well done by some super talented front end developers. My teammates were responsible for optimizing the Chromium rendering engine for that platform, to make it perform nicely on the bargain basement SoC we were using. We shipped this to millions of people and it was a success.
A few quarters later the whole thing was ripped out and replaced with Dart & Flutter. The UI was identical, but bias and and ideology dictated that that was what the UI had to be built in (they said it'd be faster: it wasn't, it was more about making it so the same apps could be built for Android and Fuchsia as the same time). Multiple quarters were spent on a transition plan to rewrite this whole UI and hack up the firmware on the system to make this possible, throwing out 2+ years of work.
So, yeah, you're right. Google was the "web" company. Now they're... I don't even know, but it's a lot about Android and mobile and also constantly chasing after what Apple, Amazon, and Facebook are doing. To the point that management was willing to throw out thousands of hours of engineering work done on a web app to replace it with a non-web version. That whole process left a bad taste in my mouth.