As an analogy-- plenty of people can play their instrument while they improvise the next bar in their head.

Try it! Transcribe the words I've written here and while thinking about the weather.

I thought most machines could do multiple things at once?

Humans barely walk and talk at the same time, and stumble much more likely than when not talking.

As someone who plays piano, you have to spend years perfecting it so it becomes automatic enough to offload your brain and only then you can improvise while playing. Your movements have migrated to a "fast" part of your brain. If you record yourself and listen to it, it's a whole different thing, you feel as if your ears were blocked to the melody, only listening enough to improvise the next key, and so on.

I am pretty sure, but to lazy to find sources which I know there are plenty, that Multitasking always yields worse quality results than doing one task. If you don't care about quality, then why don't you practice your dance moves as you write and think, as well?

If you're not doing something 100%, then you're half-assing two things.

But there are often times where I'm struck with a particularly well-crystallized thought, and typing speed becomes a huge bottleneck in recording its structure before the thought dissolves back into the ether of conceptual soup in my brain.