"Now nearly all screenreaders can run JavaScript."
Saying all screen readers can run JavaScript is like saying all graphics cards can run JavaScript. Screenreaders don't run JavaScript. Screenreaders are not web browsers. A Screenreader is an audio interface on top of an operating system. It's the web browser that runs JavaScript, not the screenreader.
I know what you are attempting to convey, despite the factual incorrectness, and the way in which you describe it. But these are tougher problems to solve in a generic way - it requires expertise on the subject matter under discussion.
I took a sentence of words at their face value, recognised the incorrectness of your statement. Applied a context of accessibility to the statement, and discovered what your intended message was (which was quite different to the statement you used).
Human beings do this all the time. A computer would interpret your statement and say it is false. A human can divine the essence of the message without being blocked by an obviously false statement, a human can look past that and assess the statement in context.
I realise, as a human, your original statement is either intellectually lazy, or misinformed (perhaps you meant to say speech browser?). You either know how screen readers work and worded the statement badly. Or you have no idea how screen readers work, and don't actually realise your statement is factually incorrect.
So I can perceive the content of your statement: Screenreaders have the ability to deal with a dynamically updating DOM. That is a step forward, but minuscule in overcoming the typical accessibility problems present for screenreader users on the web.
"What specifically else is wrong?"
"What wouldn't be solved better be actually fixing what screenreaders can't see"
Your question is misleading. It assumes or implies that all serious accessibility issues can be fixed/addressed by a screenreader. Yes, it would be fantastic when every accessibility problem is thoroughly alleviated by a screenreader. But with today's technology it can't, not even in another 10 years. Many accessibility issues have to do with content gleaned indirectly or implied. It's a human trait of communication.
It's the difference between:
* "A red triangle"
* "Danger"
* "This side is up"
* "Go this way"
* "North"
The use of images is a serious accessibility barrier for blind people. It is not sufficient (or perhaps misleading) to just describe the image in isolation. An image conveys more information than the objects/concepts in the image. The meaningful content of an image can change as the context in which the image is used changes.
Perceiving contextually relevant content from an image - that is something blind people can't do because they are blind. It is also something that's awfully tough for a computer to do reliably. And it's something that a web developer can do in five minutes by authoring an appropriate text equivalent (in an alt attribute of an image).
The big accessibility issues aren't simple "let's parse the HTML, CSS and JavaScript", they come down to using context, visual cues, audio cues, spatial cues, implied meanings, intonation, cultural identification, localisation normalisation - a wide variety of human-based factors to extract the necessary contextual meaning.
Being able to extract the appropriate text-equivalents for an image is something that's trivial for a content writer to do - as the content creator. And exceedingly tough to do without a visual acuity, a visual acuity that computers are far from achieving.
And that's just one single accessibility issue: text-equivalents for images. If you fix that, you'll have the appreciation of millions of people instantly, a substantially accessibility barrier will disappear overnight. I strongly encourage you, if you think this is an easily solvable problem, to just solve it, or help the experts solve it. Seriously. Please.