You are right about analytics, but that's an extremely easy problem to solve - simply disregard stats coming from the search engine bot. In fact both Chrome and Safari currently have a prefetch feature that I think is enabled by default, that's clearly not an issue.

For state, I don't see the connection to AMP here. RFC 7231 which defines the HTTP protocol specifically states that GET is idempotent, and should not cause any state changes in the server. The majority of servers conform to that, and if they don't, it's a bug with worse consequences than messing up prefetch.