at the very least, Google could pay to be the default search provider... it works for Mozilla, so far, and if Google had to , even at arms length, separate Chrome from the rest of Google, it would strengthen, not weaken, the case for Google's investment in paying Apple and Mozilla for top placement, helping to reinforce competition on both the mobile and browser fronts.

default search seems to be pretty lucrative.

It's data collection is in Chrome's DNA. They could just continue to collect data on browser usage, and then "sell" that data to Google. No need for sites to install Analytics into their website when they can get 100% of all site analytics viewed from within their browser.

A key difference here is that Chrome could sell aggregated analytics data for performance testing and such, but not allow the ad company to get user-targeting quality data. With them in the same company, this is a nearly impossible barrier to achieve.

I'll bite: why is this nearly impossible to achieve?

It may be nearly impossible to prove to you, but what makes it impossible, or even particularly difficult, to achieve if the company is interested in doing it?