Well, one thing I noticed in that video was how the leather was recoiling heavily with each blow. It was like one of those blow-up clown dolls kids could punch. The recoil will absorb a large amount of each blows force. So unless the person is sent flying backwards with each blow, more of each hit will be transferred to the leather. Also, None of these tests factored in the impact of the blow. Even if the blade doesn't pierce the leather, it can still bruise the tissue beneath. Particularly for those axe blows, all the weight of the axe handle/head will be concentrated in a smaller surface. Heavy plate armor from the high-middle ages wouldn't break under a hammer hit, but the bones beneath it still could.
From what I understood, the main selling point for leather armor has always been cost. Leather is obviously a renewable resource that is also a byproduct of the food-industry. Metal ores and smithing have a much more involved and labor intensive production cycle as well. The ore has to be mined, smelted, then meticulously hammered over a hot forge for WEEKS to make proper plate armor (or longer, depending on how articulated the joints are). I don't know the exact times for leather, but its no where near as labor intensive nor expensive to set up.
It was never meant to be excellent protection, but it does fair better against blunt attacks than edged ones. Against edged attacks... it is mostly for downgrading serious cuts into superficial ones, or turning slashes into deflected blows. Its the "better than nothing" protection cheaply mass produced and given to common soldiers. It should still protect against dog bites and hte like though... I mean a grizzly bear or some giant savage creature could probably tear through it (giant tigers are x32 the size of an adult dwarf), but dingos and snakes might not be able to.
DF is... weird though. A kitten can give the same leather as a hippo- both the same quantity and quality of hide. Obviously kitten leather wouldn't be enough to make a single glove, yet alone stop a sword thrust. But cured rhino hide... My point is that in the upside-down world of DF, ore is often far more abundant than leather. Also, the time to make metal armor can be roughly equivalent to hte time it takes to make leather. Thus, the main advantages of leather (cheap, abundant, easy to mass produce) are largely wasted. Instead, leather becomes this low grade armor you probably won't even bother with for a starter set... because what else are you going to make quivers, backpacks, and waterskins out of?