I don't know the neuroscience and I'm not going to pretend I do. There are studies about walking and creativity, sitting and cognition, posture and mood, but I haven't read them carefully enough to cite them with any confidence. What I can say is what I've experienced: I think differently on my feet than I do sitting down.
Seated thinking, for me, is analytical. Good for reading, for evaluating, for writing things that are already understood. Good for the execution phase of work, when the question is how to express something you've already figured out.
Standing thinking is generative. Better for making connections, for breaking out of a rut, for working through problems where the shape of the problem itself isn't clear yet. The board gets me standing. That's a not-small part of its value.
I've experimented with standing desks and walking pads for this reason. They're useful but they're not the same. Working on a screen while standing still involves a screen, and the screen brings its own affordances and distractions. The board is just a surface. There's no inbox, no notification, no other tab.
When I'm really stuck on something I do a specific thing: I push my chair away from the desk, walk to the board, and I stand there for a minute without drawing anything. I don't try to force the thinking. I just stand there and look at whatever is on the board. Usually something comes. I'm not sure why. I've stopped trying to explain it and started just doing it.