Yeah, tracking the use of screen readers isn't something that is currently available... and while the article is good and raises some valid points, I do somewhat disagree with the statement that it shouldn't be possible....
Like anything, how you use the data is where ethics come into play... as the article mentions, detection of a screen reader could be used to raise the quotes, etc... but that is an issue with how the information is being used, not the collection of the information itself... That's like saying if I track the users accessing pages with information about cancer treatments, I could raise the quotes too.... So by extended logic, I shouldn't track what pages or sections people are visiting, rendering our analytics pretty much useless...
As someone who constantly advocates for proper accessibility on our websites, my efforts are largely ignored, and I can't even help make a case for why its needed or where such users fall off in our flows. I want to track that information to make a case for website redesigns to benefit users who cannot use our sites effectively.
However, if your site has "skip links" which are a part of a proper accessible design, and are only visible to screen readers (and technically bots), you could try attaching tracking to the use of the skip links as a way to get a general sense of the traffic that is accessing those very specific functions designed to improve the experience for screen readers... and unlike adding a forced "confirm you are using a screen reader" check, this is still an optional usage, but odds are, screen readers will use that to skip your main menu and other in page annoyances during their visit. It's not 100%, but its a potential....
