How do we know this isn't just a difference in register? Classical Chinese was extremely literary, but did people actually speak that way?

This is a point I was also wondering the whole time. The vulgarisation of literature happened all over Europe at varifying times and in different stages. We don’t see these as changes in the language itself, but instead the authors daring to write the way they actually always spoke.

There’s something in bemoaning the loss of a poetic register in written language, but that’s a different and much less significant change.

That's exactly what it is. The article is almost entirely misconceived.