>> When I type, I'm able to render my thoughts directly into text, whereas when I write with a pen I have to think about the physical process of writing. I imagine I'm not the only one here who feels this way.

It is still the reverse for me. For me, writing involves less cognitive load than typing:

While writing, my mind wanders into thinking more (probably since it still has a capacity to, perhaps just because writing is slower than typing) which helps further.

While typing, I still make a lot of mistakes, which take attention away. I hate touch-screen keyboards as my error rate with them continues to be an order of magnitude higher than the same with physical keyboards even after several years of use.