You mention the [X] button. How many end users do you suppose think that the [X] on the AMP header bar should make the header bar go away...versus functioning like a back button, going all the way back to the google search? I suspect they are conditioned by things like the EU cookie warnings that an [X] on a header bar makes it go away.
An AMP page you reach from a Google search is Google's page, period. It's not the publisher's page. And that's no accident or side effect. It's the whole point. Performance is a secondary concern.