He can subpoena records if he files a lawsuit, convinces a judge that the lawsuit shouldn't be immediately dismissed and survives arguments by opposing counsel that the records should be protected from the discovery process. I.e., not "incredibly easy", as you say. Also, ISPs usually claim in response to subpoenas that they do not retain the bodies of emails and other "contents" of communications I gather.

Also, Hans (initially) didn't want chess.com to be able to publish the communications, so he might've avoided using gmail in favor of some method that is guaranteed not to leave a verifiable record.