Using the signed exchange mechanism means you allow anyone to serve your content. You will no longer know when it has been served and by whom. Instead, Google will know more about what your users are consuming on your website than you - despite HTTPS! Also, there is no mechanism to limit who is allowed to serve your content for you. I see no technical reason why the content has to be prefetched from Google instead of…
Amp does allow Google analytics and other analytics services. Unfortunately, most places don't use server logs for much :(
JS based analytics (google or otherwise) is generally a better option for detecting actual usage. Yeah, you lose maybe 2% of actual users. You also lose 99% of the various bots. You still have to filter google's and bing's bots that execute JS though.