To my understanding, "unbendable arm" is unbendable exactly because you are relaxed. Mechanically, I've heard people say that it's the exact position where the musco-skeletal structure of the arm is at its most powerful. Tensing the muscles on either the flexor or extensor side would disrupt that balance. And as to 'locking' your joint, no disrespect, but you probably misunderstood what your sensei was saying, since it's not really a full extension of the arms to absolutely lock out your joint but rather a 'stabilization' of your joint if you will - I can't really explain this.
As to extension, I always understood it as more of an outward, 'raising' movement, similar to raising a sword in an arc, rather than a 'push' per se. I think ikkyo undo and bokken suburi would probably be helpful.