I am not sure 250.104(C) implies this is a 250.52(A)(2) ground electrode although it is bonded that way.
I assume they do that because they are not defining the circuit likely to energize. If you think the 60a to the heaters in the air handler or the range is a candidate, a #10 is not unreasonable. I still think if you just used a metal box for the dryer, the range and the couple required GFCI outlets you are covered. Those are generally where your hazards are. (bath, kitchen, laundry and garage/basement if there are steel studs.)
Then pick up any wall segments you haven't hit. This is the kind of thing a decent CAD program could highlight on the plan.
