The way to divide up Google is into the ad network, the search engine, operating systems, and everything else. Allow them to contract each other on public FRAND terms to start with, but setup annual reviews to consider a bigger hammer.
If they play games with moving house to avoid jurisdiction, ban payments to and from them.
Google the search engine could charge an ad network a lot for the right to show sponsored results and provide analytics.
Android/Chrome could charge a lot for user data, analytics, have their own mobile ad network and app store. Android could also start charging a licensing fee to vendors
Etc etc
It seems like it just creates a lot of inefficiency.
For example, other ad networks can buy ad placement in google search.
There may be some completion in ad networks, most likely racing to the bottom since margins on ad-tech will be razor thin.
This will do nothing but make things worse.
Google being a search monopoly is the only issue. Breaking the ad business away won’t change anything except make things worse for the consumer.
Working out how to make search competitive is the issue.
It was always a mixture of both scenarios.
Yahoo prior to using AltaVista, Inktomi or Google had its own search tech - going back to 1995 - and its own advertising network.
AltaVista had its own search tech and its own ad network.
Lycos, a top 5 portal at the time, had its own search tech and its own ad network.
GoTo.com, very popular in its day, was both a search engine (acquired) paired up with its own famous ad system.
For several years AOL flirted on & off again with their own search tech, initially based on WebCrawler, prior to and after their deal with Excite. They also ran their own ad network.
Operating systems might have to get lumped into the 'everything else' bucket if they don't generate enough revenue through charging manufacturers or getting paid by ad networks.
Splitting up these things may not make any of the markets (search, ads, OSes, other consumer services, other enterprise/cloud services) more competitive, but it would prevent using monopoly positions in search and ads to subsidize Google products in other markets.