The point is that they were also installing tracking scripts on the website, which gives them data, which is their oil that they refine into profitable ad placement.
If you're not familiar with it, Google Analytics is a bunch of tracking scripts.
Yes, it allows you to easily verify you own your domain: in both Search Console and Analytics. Doesn't mean they'd automatically add analytics to your site.
How would they do that? If you point the DNS record to your server, the visitors call your server, not Google's. They can't just MiTM it.