In the first place, the Pedersen rifle is not gas operated, it is retarded blowback, as is the Japanese copy shown in the video. (FWIW, I have never fired the Japanese rifle, but I have fired a U.S. Pedersen; I found it easy to shoot but a bit awkward and never saw any great advantage over the M1 rifle.)
The Pedersen rilfe uses a toggle type mechanism which looks a bit like that of the Luger pistol but the Luger is a true locked breech, where the Pedersen is set up so the locking is a bit below dead center requiring extra time to move the bolt. It is simple, doing away with the complications of gas operations (especially in the days of corrosive primers), but has its own complexities, including the need for lubricated cartridges. That is not really the awful thing some writers made it out to be - the "lubrication" was hard wax, not oil dripping all over the rifle, but it was a concern if ammo was dropped in dirt or sand.
The Pedersen operating spring is not around the barrel, it is contained in the bolt assembly itself.* The use of a bolt return spring around the barrel has been used in pistols (notably the Browning 1910/1922) but has not been successfully used in a rifle because rifle barrels can get a lot hotter and ruin the spring. (Even rifles with the operating spring under (M14) or over (FN-49) the barrel had problems when the rifle was allowed to become too hot.)
Jim
*The spiral shape seen through the holes in the handguard of the Pedersen rifle is not a spring - it is spiral cooling fins machined into the barrel itself.
JK