Sounds defective.
Per the Apple support documents (including the tiny paper Getting Started document that ships with APPs), double click should be next track. Triple click is previous track. Single click is play/pause/answer call.
Note that this is the same functionality that wired Apple earbuds (the ones Apple have always included with iPhones) have had for years and years. The behavior is the same. Even Apple isn't stupid enough to change this.
This should be the same on either left or right APP. It works this way on mine.
Also a long press on the stem will cycle through the normal/transparency/active noise cancellation modes.
My AirPods Pro have firmware 2B584.