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Does Google do the same?

From my experience, Google products almost rank top in the search results without Google in the keywords.

Search "News", "Drive", "Calendar", et al.

In Google chrome, all bookmarks are under auto-completed keywords in the address bar.

No.

If you Google "Dropbox" it doesn't return Google Drive as the top result.

And if you Google "Firefox" it doesn't return Chrome at the top.

Now it might sometimes show ads for competitors, but the actual organic search results show what you're actually looking for.