I don't think I care about what brain activation happens while I'm writing per se -- though I may care about behavioral/performance differences. Do you think more deeply or remember better when taking notes by hand? Are you more likely to catch errors?

Here's one study that looked at unrelated word recall tasks, and didn't see convincing evidence of a difference between handwriting and typing. However, they leave open whether there are differences that arise in more complex learning environments. https://www.jowr.org/jowr/article/view/963/930

I think the higher-level, more realistic experiment I'd like to see someone do is: for a single college-level psychology class, split students into handwriting vs typing groups, and assure them that they'll all be graded on a curve within their assigned group, and see if top-level performance between the two groups differs over a term.

I can't solve complicated problems on a keyboard, I need a pen and paper for deep thinking. It's very rare that I go back and read any of it, means to an end.

The conclusion of that article mentions that it depends on whether your deeper cognitive centers are activated. If writing by hand, the handicap of slower transcription might force the writer to summarize or paraphrase, which uses more brain activity than if you were able to type it verbatim.

That's not to say one is superior to the other. But maybe it means that technique is more important than medium.