That would be better than nothing, and if Cloudflare had done that, I don't think I'd be complaining publicly at this point. But there are still problems with this approach. The ones that come to mind:

1. Assuming the remote service only sends down streaming audio, this doesn't work for blind people that must use a refreshable Braille display, e.g. deafblind people. Perhaps one could hack a way to get their local screen reader to render specific text on the Braille display, but probably not without that screen reader speaking the same text. That leads me to...

2. A blind user is already running a screen reader, with its own text-to-speech engine, configured the way they want it. Adding a remote screen reader to the mix would mean two different TTS engines, and the user would need to have a way of configuring the remote one, e.g. to adjust its speed. For blind people, TTS settings are very personal.

3. The remote screen reader and the local one may clash on keyboard commands. And, depending on the screen reader, this is another thing that the use may have customized already; for example, some screen readers have desktop and laptop keymaps.

4. Also speaking of keyboard commands, some of them might not be implementable in a browser-based application. It's common, at least on Windows, for screen readers to use non-standard modifier keys, e.g. Insert or Caps Lock.