Flexibility plays no role, because the rod can be naturally bent.
If the walker has no rod and his CoM is not aligned to the rope he hasn't a big chance of adjusting it.
A rod contributes to the mass of the system (and the position of CoM) and distribute mass on a greater extension, and the man can shift it sideways or rotate it. Angular momentum mvr = L of the rod is a powerful tool to adjust his CoM: he can "lean" on it and pushing it one way makes his body rotate in the opposite direction.
A heavy rod also lowers the center of mass and makes the body on the rope more stable.
The longer (r)/heavier (m) the rod, the greater is L, and the greater the push he can apply to his own body
If the rod were curved just a little below the rope (not in the above picture, of course) and had some ballast at both tips, then the walker could never fall down, if he lost his balance the ballast wpwould bring him up again.(, as long as he does not slip off the rope)
rob has misunderstood the picture and his comment has provoked the downvotes. No worries. The center of mass in the picture is much lower then the CoM of the walker, therefore the whole system is more stable.
If he carried same rod wit same inertia, but in a higher position , he would be even less stable
than he would be with no rod at all, and he would easily fall down