Latency and Inertia still exist with autocars. Anyone wishing to challenge physics may as well be awarded a Darwin Award.
But a decent brake system and tires can do 1G, which is pretty good. I suppose you could go to maybe 5G for a true emergency brake system (which would be pretty rough on the occupants, eyeballs-out), but it hardly seems worth the cost (in money, weight, maintenance, accidental triggering, etc.) compared to good sensors and the existing braking systems.
While you can instantaneously stop a saw blade to save a finger, stopping a car 'quickly' will likely move the potential problem to one of ensuring the safety of the passengers...
It would ruin the car and likely damage the road but as long as the passengers can be protected from the rapid deceleration, it just might save a life.
OTOH you could just wrap the entire car in airbags and protect pedestrians the same way you protect the passengers. Not sure how practical or useful that would be, though.
Here's a demonstration, it's very fluffy but it gets the point across. https://youtu.be/eiYoBbEZwlk
<snark>I also don't see how a car could dive below the roadway instantly. Roads don't have car-sized holes in them. Usually.</snark>
You can't play blindfolded frogger.
EDIT: "friction" is a better function name than "mu", even though we all feel nostalgic for high school physics.