Absolutely. Leave HTML as is and focus on actual accessibility features. They could even add screen reader specific elements and indicate that all other page

Absolutely. Leave HTML as is and focus on actual accessibility features. They could even add screen reader specific elements and indicate that all other page elements should be ignored. So, you would show one presentation for visual purposes and the other for audible purposes. (Or even braille for feedback purposes.) Add something to the document that tells it to ignore screen readers as long as a <screenreader> tag is present that puts everything in one place.