Even Chinese chess is played on the intersections instead of inside a grid. From my digging into their past, I feel it is more likely based on cultural reasons and some relic from precursors that influenced them, Contain metaphors that no longer needed in their later forms.
The precursors related to them might not have been played on a grid board at all, but some free-formed or various designs of graphs, and the lines connected the spots are “roads” allowing movement passage, liked in merels
I had a post discussing my thoughts and diggings A hypothetical precursor game rules of ancient Go