I certainly had to tweak HTTP user agent headers in the past to get content out of some websites. IIRC Twitter (or possibly some other website(s)?), until recently, seemed to be somewhat usable without JS, but the content was covered and made unusable with a message about JS being required -- which isn't behaviour based on reported user agent capabilities, but still intentional breaking depending on them. It's not hard to imagine that some websites will decide to not support screen readers, and to block them completely and explicitly, to avoid related issue reports and complaints (again, as they do with JS, cookies, CSS). Or at least messages like "you need to upgrade to JAWS in order to access this website", since the website will be tested just with that.

Actually I think varying behaviour depending on reported user agent was a source of frustration to users from the beginning. Maybe it'll turn out fine this time or in this case, but those concerns sound valid to me.