Wait, do regular cars engage the emergency brake in a fail-safe way? I've always treated the emergency brake as a way to keep a stopped car from moving, not as a way to make a moving car stop.
Additionally automatic transmissions have a transmission lock, but that won't work while the vehicle is in motion.
Some modern cars use electric systems for both, I'm not sure how that would work.
And the power braking system, being pneumatic IIRC, keeps working for a couple of hard stomps on the pedal even if the engine stops running and you lose 12V.
Pretty much the only thing you can expect to lose is the ABS. Even then, I understand that system has a failsafe such that it keeps the car from spinning in the event of malfunction and brake lockup. You can see this in ABS-related accidents as straight skidmarks. But I don't think that works when you've lost electric power.
Edit: actually, the recent Koenigsegg One:1 high speed crash (driver not hurt) during testing at the Nurburgring was an ABS sensor failure, you can see the hallmarks in photos. Koenigsegg also deserve big props for having been completely open about it.
And drift cars use a separate hydraulic brake attached to the rear disks.
Others (usually less expensive cars) have a set of drum brakes inside the disk brake that act as emergency brakes.
If you have rear drum brakes, it's the same as my first example. The emergency brake activates the normal braking system.