My own diagnosis is that the backflow preventor (white nylon, plus some O-rings and a washer) is not the source of the noise. There is no discharge from the backflow preventor during the noise. Now that I have opened up the American valve (from HD or Lowes, don't remember) I can see that the (nominal 12") stem is quite a complex device, which includes a spring loaded conical washer at the water control end, and a deliberately fragmented set of male threads (allmost 1/2" pipe size) through which the water flows) near the valve seat. There are also a sliding set of big rubber washers and a Bellville sort of steel washer that amount to the Bonnet Packing.
It was not easy to find a matching backflow rebuild kit. I rebuilt it myself. (I noticed that more than half of the frost-free faucets on sale at my HD had their backflow preventor stolen from the display.)
I don't blame American for the very hard water in my town. It looks like the original spring there (the moving part of the water valve) has broken, or become limed-up. $9 is not a bad price for the replacement part (which includes a pound of steel shaft.) But it does seem like a device that was designed to need maintenance. The 1955 sill cock I replaced lasted a lot longer.
Edit: And find a 3/8" 10-32 brass screw, in case the original breaks when you try to re-use the original handle.
Additional data: Because I could not remove the knob screw, I went to Lowes on 4/10/22 to buy an entire faucet. It turns out that the 12" American Valve unit has been completely redesigned between 2012 and 2022. The valve actuating threads are much coarser. As a result, the valve closes faster and feels like it may be less likely to be over-tightened. Luckily, the (solid steel) repair shaft that Amazon sent me (American Valve M72AS-STEM12) matched my "old" 2012 (plus or minus) valve.
Not only did they change the actuator thread, but to further discourage mis-use, they changed the (effective, if not actually) Bonnet Nut to a different diameter and thread, very close to the old size. So I'll have to saw off the old shaft to get the old bonnet nut.
The new (coarse actuator thread) model says "12in. M73AS", UPC 11918-11458. Made in Taiwan.