No ... that video is a rear-wheel-drive car with leaf-spring rear suspension (which is the type most prone to axle wind-up - and you can see the leaf spring in the video doing just that). It's got nothing to do with the front wheels, which are completely unconnected here.
The issue is "wheel hop". The axle winds up, the geometry of the suspension pulls the wheel upward, the wheel loses traction and spins, the axle unwinds and that tramps the wheel downward, the tire regains grip but that winds up the axle. This sets up a big oscillation, and if the amplitude of that oscillation gets big enough, you can certainly start breaking things. You can see the axle hopping and tramping up and down together with the leaf springs winding up and unloading in that video.
All sorts of add-ons have been invented to combat this, usually called "traction bars" or "slapper bars" or some such thing.
Other types of rear suspensions which don't have as much compliance in the axle-twisting direction are the proper way to fix this.