No. It doesn't. Components are rated in force, not energy.
F = M*A. The stretch and time of the strap massively reduces the A
The reason shit goes flying more than it used to is because all that kinetic energy is momentarily getting stored in the rope which is acting as a spring so when you do put a few ten thousand pounds of tension on it and break something you've now got all that stored energy in the spring that is gonna find a way out. The straps back in the day weren't storing all the kinetic energy of a (for example) 10mph down to 2mph acceleration even if they were subject to more force, say 5mph down to 1mph over a tenth of less of the time, that energy mostly went into breaking shit, usually breaking the friction between the stuck vehicle and whatever it was in but sometimes something else.
The pulls leading up to it were even harder. It wouldn't surprise me if that first one in the forward direction was the one that did the bumper and the soft shackle didn't fuck off until the bolt head was hard against the plate and the soft shackle became the new weak link.
Alright I will join the shitshow even though none of you asked.
This is going to be a long drawn out response.
You got the right idea but your physics is wrong. The amount of force needed to get unstuck does not change, there is a certain amount of restrictive forces pulling back on the bronco keeping it stuck that have a maximum value. Once you overcome that force it will move, and it will not increase (although once in motion the force will cause it to accelerate). Draw a simple free body diagram and it will make sense. Go and push on the wall, nothing is moving, yet there is still a force between your hand and the wall. (Newton's third law). The force (in this case tension) in the rope has to reach a certain amount before it will move the stuck vehicle, regardless if it is a strap, chain, or rope, or how fast the vehicle accelerated.
If we treat the rope like a spring we can use the equation F=Kx, and energy stored in a spring uses E=1/2Kx^2. K is the spring constant for the rope, x is distance stretched, F is the force required to move the vehicle, and E is energy sent to the system (Kinetic energy from the recovery vehicle, or work produced by the distance the vehicle moved.) Solve the energy equation for x and plug it into the force equation and you end up with F=(2KE)^1/2. Solve for E we get E=(F^2)/(2K). This tells us how much energy is needed to produce the force to move the vehicle, and any other energy goes to two places, either momentum that is what gives the stuck vehicle forward velocity, or into deformation, which in the case of a rope is absorbed into the rope. That rope, since it is like a spring, also will want to return to its original shape applying a force to the vehicle helping to accelerate it through whatever else just beyond what it was stuck in, and since you are moving at a slower velocity it will be nice and gentle.
The K of a chain or strap is very large, and it will not deform much. You do not need the same amount of energy to apply the same force as a rope. But if you do that energy either has to go someplace, usually into faster velocities due to momentum (the massive jerk forward that gives you whiplash) or into deforming the recovery points/the vehicle instead of the rope/chain, which can lead to failure of the parts. You also do not get the same gentle pull after the fact.
Now the disadvantage of a rope, like you said, is it stores a ton of energy. So if it fails, it will fail pretty violently. A strap will not store as much energy, so it will fail less violently.
Keep the equipment in good shape, use it as designed, and it won't kill you. Did not watch the video but I think the tow points/frayed shackles are what caused the failure, so both of those rules were broken.