This would be incredibly useful and helpful.

Personally, I believe that turning live regions on and off should be controlled by the Web developer, not by the user. Authors expect live regions to be announced. If we add a setting to turn them off, or worse, if we add several settings to adjust how and what they will announce, web developers will have nothing to rely on any more.
Random users will start facing issues with random web apps, like Google Docs, just because they changed a setting at some point and then forgot about it.
I have lots of experience with other assistive software that has similar settings and I have met several users who were in a situation like the above, i.e. they changed a setting and then forgot about it and even had to pay for someone to fix their computer.
Not everyone has the time or the inclination to become an expert in their screen reader's settings.
Also, the ARIA Standard says that live regions should be  announced with a specific priority or politeness level, and I think that screen readers should respect that. Authors will expect this behavior and some authors will even rely on it. How will authors reliably test if on different computers there could potentially be different behaviors? Do we expect authors to learn both the ARIA Standard and the settings of all the screen readers?

By the way, are there specific websites where live regions have been too distracting so that they had to be turned off?