When I type, it's effortless and I can type almost as fast as I can think -- so I'm using my brain efficiently.
When I write by hand, I hate it. I have to pay attention to how I'm holding the pen, if it's legible, if I have room for the next word on the line, if I'm going to smudge something, if I can keep up with what's being said or just cross this sentence out and write the next one before I forget that one too.
So sure, my brain is way more engaged when I'm writing by hand. But it's engaged in a bunch of BS related to writing, not the subject at hand. If anything, it's making it harder for me to pay attention, which (at least for me) is definitely detrimental.