Not really sure what they were trying to get across there but I'll give it a shot.
These videos don't really explain of much of why things happen but they do demonstrate a little that is helpful. First, people complain about understeer but it is is designed into vehicles because the majority of people can't handle oversteer. When the tail kicks out, they freak out, grab a hand full of steering and jam on the brakes as hard as they can. Bam, instant loss of control. 4wd can help overcome that on loose or slick surfaces.
You can argue that you don't need it 4wd while driving faster than you should on loose surfaces but that will only go so far. If you are driving too fast in dry dirt, yes, you can recover and be fine. In snow, ice and slick mud, the extra control 4wd offers can keep you from crashing by allowing all the tires to have some input on speed and vehicle control. 4wd won't stop you any faster but it will help directional control immensely in those situations where no matter how slow you go, there isn't enough traction to allow you to follow the road or to avoid obstacles.
That is where adding power and steering where you want to go comes in. You are no longer along for the ride at that point, you are doing something about it. The old adage of leaving it in 2wd until you can't go further or absolutely need it is false economy and can really hang you out to dry unnecessarily. I've driven a lot in loose and/or slick conditions and used to believe in the 2wd theory until I began driving fast for a living in poor conditions. Having the ability to steer where you want to go and accelerate sure beats letting off, counter-steering and hoping the tail doesn't pass you due to inertia overcoming traction. AWD won't help you much here without traction control because it will transfer all the power to the tires with the least amount of traction. That is usually the tires that are sliding sideways and that weight transfer has lifted from the ground or unloaded severely.
On the subject of AWD vs. 4wd they both have their places but in an AWD vehicle, you can have all kinds of issues if you are not used to driving a car that will transfer all the power to the wheel with the least amount of traction. If you turn a car sideways, that will be the tire that is trailing and has the least amount of weight on it. Unfortunately, it will do you almost no good to add power to that tire since it won't have enough traction to do much more than emit smoke or sling loose surface material around unless you have some sort of electronic traction control.
Driving habits are also part of the equation. If you are used to a front wheel drive car, it will almost always understeer or "Plow" until enough friction builds heat into the contact patch or in dirt, until the speed drops enough to allow the available traction to overcome the vehicle's desire to go straight. This is due to the design and the weight bias. A rear wheel drive car can induce over-steer by throttle on or throttle off. AWD vehicles can have the worst of both worlds built in and a 4wd vehicle can have the best of both worlds, or the worst, depending on the surface and driver inputs.
As for the trailer show. Pulling a trailer increases over-steer because the trailer is pushing the tongue in a straight line. The truck is trying to turn, the trailer wants to go straight. Being in 4wd allows the driver to accelerate the truck and pull it back straight. If you've never lost it in a corner with a trailer, you won't fully grasp the importance of having a manual way to activate trailer brakes ready at your fingertips or the control that 4wd will bring to that situation. 4wd also forces the front tires to keep up with the rears and vice-versa. Letting off the throttle can cause the rear tires to stop rolling and start sliding sideways and as this happens, the front inside tire can stop rolling as well; causing big problems that cannot be overcome with talent. 4wd causes all the tires to at least want to keep moving, absent differential action.
On the subject of recovering from a slide, although I'm not a huge fan of auto transmissions; there is merit in being able to grab the trailer brakes and stab the throttle without having to gear down to get enough power to do the job.