Depends on the job.
I'm currently hiring and the live coding exercise is a must for the position because I want to see how the person problem solves while under stress.
Because stuff happens. My team works on mission critical systems that can have issues that we sometimes must resolve quickly which is very stressful.
If you can't handle the stress of me watching over your shoulder while you code then you can't handle the stress of getting a critical bug fixed immediately. Maybe that's not true for everyone but I haven't heard of a better filter.
Not every position is like this. But that's the big thing I think people miss. Different software positions require different skillets beyond just tech stacks. And the interview process should measure for the particular software engineer skills needed for that position.
I've also hired people who if you just reviewed their code after the exercise you'd think they were terrible. But I'm not testing whether they are good at coding exercises, the test is a tool to see their ability to problem solve under pressure and to see their level of experience with the particularly tech stack.