On page 107 of the manual:
"WARNING: Automatic Emergency Braking is
not designed to prevent a collision, but to minimize the impact of a frontal collision by attempting to reduce your driving speed. Depending solely on Automatic Emergency Braking to avoid a collision can result in serious injury or death."
Thus, when I drive at 40 mph and there are stopped cars at a red light, if I rely on the Automatic Emergency Braking to prevent a collision, most likely, it will only brake to slow down and not brake to avoid a collision. That's what its designed to do.
Lucid Adaptive Cruise Control doesn't brake to avoid a collision for the above scenario either. However, if you approach at a slower speed like 20 mph, your Lucid Adaptive Cruise Control can stop to avoid a collision.
Your automatic braking for cross-traffic works because, the speed of your car is really slow, like zero or a few miles per hour.
I suspect as Lucid will write programs for the LIDAR, collision avoidance will be better and the system will be able to brake to a stop to avoid a collision.
When you said "smart car," it helps if you know what level of autonomous driving is. Lucid is classified as 2 which means with that level, a human is still require to bail it out of collisions.
Mercedes has come out with L3 Drive Pilot and it can brake to avoid collisions but notice at its speed: at the maximum of 40 MPH. That speed will be increased to 80 MPH but not in a near future.
The ability to brake to avoid a collision with a stationary object depends on hardware and software.
Lucid got the hardware LIDAR, but it needs software engineers to work on the function.