I believe written assessments were relatively new to standardized tests when I was in the field. My understanding was that the purpose was to assess writing, just like we do with, say, math or reading comprehension. Multiple-choice questions can't evaluate a student's ability to respond to an open-ended question, convey ideas, stay on topic, demonstrate grammar and punctuation, etc.
But I didn't get the impression that adding writing components to the test added anything of practical value. Sure, it's one more score-based judgment that can be assigned to each student and be used to assess schools and districts, but as far as I could tell it wasn't benefitting student education. And it certainly came with a high price tag. But I was definitely not in classrooms, so that's a key caveat to anything I share.
Does writing seem any better? Well, these written assessments have been widespread for about 20 years or so. I don't get the sense from what I see that people nowadays are generally good writers. But my exposure is certainly limited, even if others I speak to seem to agree, and I'm not certain if enough time has elapsed whereby we would broadly see the effects of such a change in educational assessment.