I mapped out the structure of the board as a graph of which squares you can get to from which other squares. It's surprisingly stringy without all the squares forbidden by the queen placement.
Hah, you beat me to it! I drew this very same board by hand, after noticing that many positions were dead ends, and that the options for moving around the top-left were quite limited.
Thanks for generating this! I struggled quite a bit getting to a few squares, and the graph shows they were indeed far apart from each other and that I wasn't missing something obvious.
This is a good advertisement for the approach I came up with eventually: if you don't see a trivial way to the next square, find a sequence that will lead to it that starts in the lower right quadrant, then go to that quadrant and get to that square. (The lower right quadrant is the center ring in this diagram.)
I'm used to a version with pawns that involves less backtracking, so it took me a little while to stop trying to be "smarter" and just be okay with continually returning to home base.
https://ibb.co/JFY3jm4
It made a lot more sense after doing that, as I'm nowhere near proficient enough at chess to see the board and all its moves in my mind's eye.