I do not know why but I did not find Chambers Bay all that challenging to walk...but maybe I was having too much fun playing! Ashamed to say I carted Highland Links....but I reckon that would be a gruelling hike. Australia's Hamilton Island (or Dent Island, if you prefer) is another that I carted and cannot imagine walking.
That said, Old Course at The National (Australia) would get my vote...especially as (I am guessing) 80% of the members walk it. Carting is not as common down here, especially for private club member play. Seniors, ladies....everyone walks it, some , like myself with a bag over their shoulder. Its actually far easier to walk & carry than trying to push or steer a trolley around there.