I don’t think it is just a privacy issue with regards to data but also the fact that visually impaired users fear they will lose their right to be treated like anyone else.
The web is a place where nobody knows they are disabled and are therefore not treated any differently, something that unfortunately does not happen in the real world.
As a developer I understand that having information on who uses screen reading software could be an advantage with regards to proving to product owners that we need to account for these users, in the same way we do for IE users with conditional style sheets.
I can see both sides of the argument but the solution is simple, developers (myself included) need to code sites with the visually impaired in mind. This needs to be ingrained into devs when they are learning to code. Just as we learn to indent code, use nav tags for navigation, we should be taught WAI-ARIA markup and keyboard navigation.
Sadly as far as I am aware this isn’t happening.