Except that officially, that is not what they do? It's a dick move not to put the two usecases under separate user agents, but their documentation says you're free to block Google-Extended via robots.txt which is used for training and grounding, while still being included in the search index.

> Google-Extended does not impact a site's inclusion in Google Search nor is it used as a ranking signal in Google Search. https://developers.google.com/crawling/docs/crawlers-fetcher...

Exclusion from grounding does mean that your site won't get sourced in the AI overview, but I'm not sure what the click through rates are like on those.