You do not, but that wasn't the hypothetical posited by OP.

In the actual case with Tesla, the driver of the car that was in front of the pile-up was not at fault IMO. But there were a bunch more cars behind, and at least some of those slammed into each other because they didn't maintain proper distance, according to the police.

You don't, but as soon as you see the car in front of you change the lane (or just indicating via turn signal), you should? And to be honest, this looks more like slowing down and not a hard break, and at least the first following driver slept quite a bit... agreed the stopping had no reason, so (as someone quoted German laws) for that you would be too blame... but for crashing into the car with that video footage, I'm sure the following driver would get at least 80% if not 100% blame.