If you watch the video again carefully, the Tesla alerts before the car in front brakes, because the car in front of that is braking.
"Skoda Front Assistant is a system designed to help avoid or to mitigate accidents into the rear of preceding traffic. A long-range radar, positioned at the front of the car, can detect vehicles up to 80m ahead which the car is likely to hit unless action is taken."
Keep in mind, Skoda is not an upscale brand, they are on the budget end of things. The VW and Audi models might/should have better systems.
The new Peugeot 308 also offers a 'Emergency Collision Autonomous Braking System' (http://www.peugeot.co.uk/showroom/308/5-door/p=safety/) but on closer look this is actually 'an automatic brake application to reduce speed by a maximum of 12mph to reduce the severity of impact'. Much less useful, but at least moving in the right direction.
One problem with suddenly braking hard to avoid a collision in front is then you get rear ended (happened to me). Sometimes the best approach is to swerve onto the shoulder instead (also happened to me, and I watched the guy behind me plow into the guy in front).