The divisions are arbitrary.

A search engine is a number of components - e.g. a crawler, an index, and a user interface.

Why should one company be able to own all three of these? Why not just one? Perhaps crawling should only be able to be sold as a wholesale commodity.

What is the exact point that a crawler needs to be decoupled (at the corporate level) from it's front end? Is it measured by MAUs? Pages scraped? Or is it going by "it can be decoupled, so it should".

The extreme version of your argument would have companies own the IP to a particular for loop's iterator variable name.

Indeed.

“My” argument is a generalization of the argument people are making that Google should only be allowed to operate “one” service.

So the ad business, which generates all the revenue is split off from everything else. All the various services including search die b/c they can't self sustain and all that's left is an ad platform akin to the former double click scenario. Now what?

Bing becomes the number one search engine.