I think you're assuming Chrome's very existence was inevitable, but it wasn't.
Chrome was created partly because Page just wanted to do a browser, but that wasn't enough by itself. Schmidt blocked a browser project for years on the grounds that it was a low priority and there wasn't much reason because Firefox was doing fine.
What changed things was Google trying to work with Mozilla to contribute resources, push the web forward faster and running into huge problems again and again. Political, corporate, technical. The Firefox architecture was over-engineered but also the Firefox guys thought they were top dog and kept dropping or ignoring Google's contributions for what looked like spurious reasons. Frustration grew amongst engineers who cared about improving HTML and with Page having always wanted to do a browser, now there was pressure from the top and the grassroots. When they got a few key hires who showed they could do a new browser with a much better architecture, and move way faster than cooperating with Mozilla, events were set in motion.
But there's a parallel universe in which Mozilla welcomed the Google contributions with open arms, in which Gecko had been written in normal C++, was easier to work on/better documented, where the app architecture was more conventional and thus more amenable to sandboxing etc. And in that world maybe Firefox would be even more dominant than Chrome. I don't think Mozilla realised back then they either could learn how to work with the Google engineering teams, even to the extent of giving up some architectural control, or Google would wipe them out.