I love a bit of experimental archaeology, but there are a few caveats to this video.

1. The crossbow he's using replicates the power of an English longbow in good condition. ie, the upper end of bow power that was achievable in the later Middle Ages
2. He is firing at very close range, and at that point most archers would be thinking of drawing their swords or bravely running away. These bows were designed to be used at a range of up to 300m, as we know from surviving regulations on archery practise that mandated proficiency at a minimum range of 200m, and from reconstructions that show 2-300m to be the maximum range of late medieval English longbows, and from late medieval/renaissance records of archery contests where hitting a target 300m was considered 'good'. Of course, they could fire closer than that and often did so, but being too close opens up the archer to being caught out of position. Firing at a greater range means the arrow will lose some of its energy by the time it hits the shield. Accounts from the high middle ages, when English longbow tactics were yet to be formed, routinely refer to soldiers looking like hedgehogs or porcupines from all the arrows sticking out of them, but they would be fine.
3. As you can see from the video, it really depends on the arrow being used. The needle shaped bodkin arrows would indeed be seriously dangerous, broadheads not so much.

So to your question of 'why not stick a metal plate on the back?' After all, having an extra layer of metal could really help absorb the impact of arrows. But it also hinders the user in a few crucial ways. The main problem is that it would make the shield significantly heavier, which in turn makes it more difficult to use over a prolonged period of time or in reflexive actions (ie positioning the shield to protect from an unexpected incoming arrow).

The second issue is that it's a bit superfluous. Sure, having arrows go through an inch or two would seriously harm the user's arm if they weren't wearing much armour on it, but front line infantry were already covered in protective equipment. The best equipped would be wearing plate armour vambraces over a suit of mail, which was in turn worn over the gambeson, a dense fabric coat. More lightly equipped infantry would still be wearing the gambeson and some form of vambrace, though it may be made from leather or splintered leather (leather reinforced with strips or rods of metal) rather than metal. In other words, even if an arrow gets a couple of inches past the shield, it wouldn't have the energy to get through the armour of the person beneath unless it was a particularly nasty arrow fired at close range (as in the video). Therefore, even without any metal reinforcement placed to protect the user's arm, the user's arm was already going to be basically fine. It would hurt, of course, but nothing life threatening. By the time the English longbow hit its peak as a weapon of war, soldiers were gradually moving away from the heater shield because its additional protection was seen as rather unnecessary when most knights were already decked head to toe in plate armour. Shields reduced in size as their purpose moved to blocking specific attacks (which required greater freedom of movement for the user) rather than offering general protection to the user's body.

So in conclusion, adding a plate on the back offered little substantial advantage whilst limiting the user's ability to move the shield to respond to incoming threats over the course of a battle. The video is looking at a slightly unusual situation, where a longbowman would be firing at very close range with armour piercing arrows. This did occasionally happen and it could be devastating, but it was not the norm.

HOWEVER! Some medieval soldiers did think the extra protection was worth the loss of mobility. Some heater shields have been found with metal plates in their construction, just not confined to a back plate protecting the arm. Instead, a thin layer of iron or steel was incorporated into the entire shield. Not many though - it doesn't seem to have caught on. A layer of steel would have added a couple of kilograms to the weight, and that's a lot to haul around a battlefield.