HAPPY TO REPORT BACK.
I have an answer.
Turns out the flappy thing is called a “trail rudder” and is very much a useful addition: It (as someone here already guessed) helps with control running downhill. It’s akin to a “snowshoe” effect, but just on a much smaller scale. It seems to primarily help with downhill angles and prevents ankle rolling or heel slippage. Which, is easy to imagine in your head.
It also offers (again, as someone upthread guessed) more surface area for more traction in general, kind of like fatter swamp tires on a truck.
Shoe companies say that this helps with the heel-strike and use it in conjunction with other trail-running features such as a wider toe box so your toes can splay more naturally with each running strike. Hoka has a proprietary term for theirs, and “trail rudder” seems attributed to Altra, but also in general.