I think that the video has a confusing message. Perhaps he is reacting to those who say that all bullets will start tumbling immediately after hitting the transonic zone. Most bullets don't, but what actually happens is complicated and affected by the shape of the bullet and the spin in addition to atmospheric conditions. Yes, crosswind and other atmospheric conditions make it difficult to hit small targets at extreme range, but the presenter notes that accuracy tends to get significantly worse as the bullet pass the transonic region. He does not go into much detail, but acknowledges that some bullet shapes are less effected than others by the change to subsonic flight. The round nose 45-70 he noted is a good example, but there are current designs, like the Berger Juggernaut, that are specifically designed to stay accurate as they transition to subsonic speed.
As noted by several people already, spinning the bullet faster generally makes its trajectory more consistent as it approaches the transonic region. Today's high-quality bullets can tolerate the greater spin rate and provide consistent accuracy. This is helped by a good chamber and throat design that make certain that the bullets are correctly lined up as they pass through the barrel.
One of the main reasons the rifling twist has historically remained slow has to do with the relatively poor quality of bullets in the past. There are several issues with bullet strength and construction, but an interesting one has to do with bullet balance. A bullet that is not perfectly balanced along its long axis will try to spin off center as it travels down the barrel. The barrel will mostly keep that from happening, but as soon as the bullet leaves the muzzle, it will tend to jump to one side, affecting accuracy. It is even possible for an unbalanced bullet to maintain a stable wobble as it passes through the air, flying with higher drag and again, making the trajectory inconsistent from bullet to bullet. Over a hundred years ago, Dr. Mann (inventor of the "Mann barrel" used for accuracy testing) deliberately unbalanced bullets to see this effect. The point of impact varied with the orientation of the heavier section of the bullet as it left the muzzle. Slow twists helped minimize the effects of unbalanced bullets on accuracy.
As noted by several people already, spinning the bullet faster generally makes its trajectory more consistent as it approaches the transonic region. Today's high-quality bullets can tolerate the greater spin rate and provide consistent accuracy. This is helped by a good chamber and throat design that make certain that the bullets are correctly lined up as they pass through the barrel.
One of the main reasons the rifling twist has historically remained slow has to do with the relatively poor quality of bullets in the past. There are several issues with bullet strength and construction, but an interesting one has to do with bullet balance. A bullet that is not perfectly balanced along its long axis will try to spin off center as it travels down the barrel. The barrel will mostly keep that from happening, but as soon as the bullet leaves the muzzle, it will tend to jump to one side, affecting accuracy. It is even possible for an unbalanced bullet to maintain a stable wobble as it passes through the air, flying with higher drag and again, making the trajectory inconsistent from bullet to bullet. Over a hundred years ago, Dr. Mann (inventor of the "Mann barrel" used for accuracy testing) deliberately unbalanced bullets to see this effect. The point of impact varied with the orientation of the heavier section of the bullet as it left the muzzle. Slow twists helped minimize the effects of unbalanced bullets on accuracy.