The short deadlines are because neither the company nor the candidate wants to spend a month on an extended interview. Solving a problem and walking through the thought process are because that's what "coding" is.
So says the companies that insist on multi-round, multi-week interview loops.
Like I said the deadlines work for both sides. If a company wants to give homework instead of having their own senior engineers spend time talking to me, that tells me what I need to know about how they value my time.
That's not equivalent to what I said, nor is it live coding.
Again, those deadlines are artificially short compared to real world scenarios, and completely arbitrary. They are so short, in fact, that they render the interview an invalid test of real working ability. A work sample has been proven time and again to be the most valid measure of whether a candidate can actually perform the job, but the conditions under which a live coded "work sample" is performed in an interview render it invalid.