I used to work at Google, so let's look at the practical difficulties with this idea of splitting. It's not impossible - AdSense exists after all - but it'd create a lot of problems.
Firstly, it's not simply about drawing some arbitrary lines between search and AdWords and saying, right, split done. Other companies have to be able to compete on equal footing. Today Google is one company with a unified infrastructure for everything. It's not a bunch of unrelated businesses loosely strapped together as Warren seems to be imagining. For instance "undoing" the merger of DoubleClick would be impossible as DoubleClick doesn't exist anymore, in any form. One of the first things Google did after the acquisition was reinterview everyone and fire about half the staff. Their tech infra is long gone, etc.
So what does that mean in practice. Well, the ad auction has to run physically very close to the search engines, because search responds very fast which means ads must also run very fast. This would be hard for other firms to do well because Google datacenters are often built in the middle of nowhere for various reasons. If a search has to hit a Google DC then go back out and across America to some other DC run by some other ad firm, then search latency will go up and people will search less, or more likely, those ads will just miss their deadline more often, won't run, and lose out to son-of-AdWords which is faster.
Could competing firms run colocated in Google datacenters? Maybe. Google Cloud exists. But you can't choose in Google Cloud to run in space allocated next to existing Google services, in fact their locations constantly change and isn't public. Ads datasets and processing requirements can be very large. If a building is full, and a new ad company says "to be competitive we must be physically close", that means something else has to move out to make space. To what extent is Google allowed to say, no, we need that space to run Maps servers or something like that? How often can Google force co-located servers to move? Does a regulator start micro-managing Google's datacenter space?
Now what about the rest of the integration? Clickfraud detection involves deep integration between web search and the ads system. Is Google obliged to make all that infrastructure on the web side available to any ad firm that requests it? What stops me creating a trivial "legit" ad firm, obtaining all of Google's secrets and then selling them to click fraudsters for massive profit?
In theory all these problems can be solved, but likely only by creating constant political fights, everywhere, all the time. And for what? Warren is engaged in leftist populism: the vast majority of people in the real world don't care about splitting tech firms up like that. Even in anti-US-tech Europe, polls show people's top concerns are usually immigration and terrorism. "Can other firms compete in the AdWords auction" doesn't appear in the list at all, it's a non issue.