If you held a gun to my head and told me to build a Google competitor, I'd start off with smaller products with shared search infrastructure.
Start off with some hyper-focused projects that you can actually compete with Google for: code search, product search, news article search, review search, search focusing on kids, etc. Build a better YouTube search and pay a legal team to force Google to let you have some access to it. Maybe a better and free "search my site" functionality. Make some of them good enough the people just use them, get Bing or DDG interested in supplimenting their results with yours.
Have these as separate brands that people know, but for every search have a button that takes you to the wider search and start building brand recognition.