It sounds like you might mean something different by emergency brake than I think most non-Tesla drivers expect. I've always called the lever/pedal with a direct mechanical link to the brakes the emergency brake. Is this a thing on Teslas?
One of its functions is to allow you to slow down and maybe stop if the hydraulics fail. It's also called a parking brake, but that's an anachronism for cars with automatic since they have a parking mechanism. Using it to park is still a good idea to keep it functional so it's there in an emergency, and in case the parking mechanism fails. Some people find this out the hard way!
I don’t know about modern cars, but people used to say that parking an automatic on a slope without the parking/emergency brake would damage the transmission’s “parking pin” and make the “park” gear equivalent to “neutral”. No idea how true that is.
Bigger issue is the human in the loop. If you always engage the parking brake when leaving the vehicle so that it becomes a reflex then you never forget. People will claim that this will never happen to them and that it is pointless to do on flat ground... Until one day they hop out of the vehicle with the engine running / key in ignition and not in park and it rolls away from them.
Next biggest issue is probably if the car gets hit hard enough to break the parking pawl. Parking brake could save it from rolling away.
The pawl itself is not a tiny pin and will take some abuse, but maybe if you talk to mechanics in SF where people park on hills they see them fail all the time? That isn't the primary reason why I'd suggest always using the parking brake, even on flat ground though.
There's the "Emergency brake" (aka e-brake) which is a misnomer for what is actually the parking brake (not to be confused with the Park setting in an automatic transmission) or also called the hand brake for cars where this brake is a hand lever rather than an extra foot pedal.
And then there's "automatic emergency braking", which is a computer-controlled automatic brake that gets applied when it detects an obstacle.
Words take on new definitions all the time, but you already saw multiple people in here (me included) assume you meant the original and much older meaning. It's good to define new and unconventional uses when it's not clear from context, and it isn't here if (like most people) you've never driven a car with this function.
It's also how I learned driving safety. Our safety instructor would make us (slowly) drive over a wet rubber road surface. He also made us evade an object, randomly having to go around it to the left or right. And then one time he'd just pull the handbrake without telling us while we tried to evade.
