While none of us here can say for sure, I wonder if this is more forced by Amazon's fleet insurer raising rates rather than an evil directly from Amazon just because they can.
Amazon has roughly 30,000+ delivery vehicles. I don't know their accident rate. But any injury or accident lawyer can tell you that listening to music and singing and generally be-bopping down the road IS distracted driving. This will cause larger payouts per accident and also higher insurance rates. And it's low hanging fruit to go after in court. If Amazon is noticing the extra cost becoming an issue, it could well be the reason for such a draconian policy.
I'm not sayin'. I'm just sayin' it could be a driver, (pun intended), in this instance.
That's not how insurance works. They monitor average speed, acceleration, and braking, if anything. There's no correlation between mouth movement and accidents. What if someone is chewing gum? What about adjusting dentures? What about drinking coffee while parked?
Show me an actuarial table that includes "mouth movement" as a variable or admit this is just a middle manager trying something stupid.