An example of the type of business relationship Warren probably wants to encourage: the way Google pays Mozilla to keep Google Search as Firefox's default search provider.
So, if Google were split up into Google-the-search-engine and Google-the-ad-agency, Google Search's main source of revenue would still be from ads—but it'd be via the separate company of Google Ads, paying Google Search for ad placement.
This would almost certainly lead to less privacy. Google Search, instead of just sourcing ads from Google Ads would have to open bidding to a wide variety of ad networks.
That means your personal search history wouldn't just be accessed by Google, it would be publicly available to any upstart ad network that wanted to bid for Search ad placement.