detect when [ADA] software is in use?
There is no known standard. Each "reader" vendor may send hints via HTTP header variables, but there is no guarantee they will be the same on the next version.
Plus, mirroring all the "regular" content with an ADA version is a bear. Authors would have to be diligent to keep them in sync.
Maybe if we had a clean CMS it may be possible to simply generate the appropriate content format from a single set of content (data), similar to some mobile-friendly presentation techniques, but right now we have a hodge-podge and historical baggage. Plus, if authors don't use the editor it right, it can still be out of whack, such as "fake" indenting of outlines.
Is it possible to provide an accessibility specific phone number...
We considered that, but it would probably have to be staffed 24/7 to match the website's availability. In other words, if we offer 24/7 service to "regular" readers, we must do the same for the sight-impaired callers, otherwise they could claim discrimination. It's a big org such that no one help-desk person will know everything, meaning you'd have to pay specialists to sit at the phone desk at 3am every night.
All known solutions require lots of resources we don't readily have. Management keeps pressing for an easy way out and easy, cheap answers, but I cannot give any.
Excellent questions, though.