I totally agree with your apprehension
I recall a restaurant that I used to eat at in University and they had a large two-layer metal grid of round metal rods that all of the lighting was screwed to. One day I was staring up at them and realized that there was no power running to the lights at all and that they must be getting power from the grid itself; that's when I realized that it was a two-layer grid and... well then it all made sense. Point being, there is some precedent that this is a feasible approach.
Aesthetic concerns aside, would it be possible to decouple the physical mounting and the power flow? Use whatever mechanical means to mount the board to the rods, and then a separate wire that clamps onto the rod?
[I just remembered that I have some lights downstairs that have a similarish mounting system as what you initially described, I'll post a few pictures once I figure out how to get them off my phone and into the forum]