> It provided one-click access to setup website, Google workspace, provided direct verification in Google Search Console and Google Analytics, and just made sense.
If you're not familiar with it, Google Analytics is a bunch of tracking scripts.
> provided direct verification in Google Search Console and Google Analytics
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.
If you're not familiar with it, Google Analytics is a bunch of tracking scripts.