To me the Code is clear, 250.66(A) addresses the "size of alternating-current grounding electrode conductors (Section title) to (A) rod, pipe or plate electrodes is not required to be larger than #6.
That section does not even discuss routing simply provides the size.
The only caveat is "sole connection to the electrode" which covers the somewhat unique situation where one might run to the rod and the from the rod to another electrode such as the water system in which case the #6 would be a "weak link in the chain", but as long at the electrode conductor in question attaches only to the ground rod it needs only to be a #6.
That section does not even discuss routing simply provides the size.
The only caveat is "sole connection to the electrode" which covers the somewhat unique situation where one might run to the rod and the from the rod to another electrode such as the water system in which case the #6 would be a "weak link in the chain", but as long at the electrode conductor in question attaches only to the ground rod it needs only to be a #6.