Saigo Shiro was the adopted son of Saigo Tanomo. It is rumored that he was actually Tanomo's illegitimate son, but that has not been verified. Saigo Tanomo was Takeda Sokaku's primary instructor, in addition to his father Takeda Sokichi. It is from Tanomo that Sokaku inherited the art of Daito Ryu Aiki Jujutsu. There is some controversy as to whether or not the art taught by Sokaku was called Daito Ryu prior to his teachings, but I will not go into that. Whether or not the specific name has been transmitted through the generations is not as important as the material, which has a verifiable lineage to Minamoto Yoshimitsu (1045-1127).
Prior to Takeda Sokaku inheriting the art from Saigo Tanomo, Saigo Shiro was supposed to be Tanomo's successor. However, Shiro was involved with a highly nationalistic group of people of which Kano Jigoro was also affiliated, and he turned away from Daito Ryu to pursue the new art of judo.
Witnesses to Saigo Shiro's matches did not have his internal background and therefore could not accurately describe or replicate his movements. Tomita Tsuneo described the technique in his book, Sugata Sanshiro. In it, he says "The technique of yama arashi is not depended upon powerful arms or body. It depends upon a strong mind, such as, the spirit of go for broke. According to the dynamics, the logic of yama arashi is based on the breaking of the opponent's balance within a short distance and attacking with full speed. It is the most dynamic and skillful of all other techniques. If we hold each other in right handed position (Saigo was left handed), Tori grabs Uke's right collar deeply with the right hand and grip Uke's right sleeve with the left hand to take an extremely right handed position. Tori moves his body up and down to pressure Uke backward. In response to Tori's controlling movement, Uke tries to push forward to regain his position. When Uke tries to come forward, Tori picks up Uke's body on his shoulder in full speed and sweeps Uke's right ankle like a Harai goshi (a gust of wind). It can be considered a combination technique of Harai goshi and Seoi nage."
However, if you look at the statue of Saigo Shiro in the upper left corner performing yama arashi, the sweeping motion mentioned by Tomita isn't there. It's as if the attacker is being thrown simply at the shoulder. Despite Shiro calling this technique yama arashi, it is actually a traditional Daito Ryu technique called kuruma daoshi. Upon contact, a rise is created (aiki age) while projecting uke's force to one side of their body. When they push back to try and compensate, there's a slight rotation and drop (aiki sage). Without understanding or having refined the internal principles of aiki, this would be a very hard throw to pull off against a resisting opponent. You'll also notice that in the statue, uke is falling on his head, characteristic of the combative application of koryu jujutsu opposed to the modified judo variations. For Saigo Shiro, a man being trained to inherit the art of Daito Ryu, it's simply another day at the office.
But why did Shiro change the name of the technique? Why wouldn't he want to give credit to its origins? Prior to Takeda Sokaku taking the art public, the techniques of Daito Ryu and the aiki principles that drove them were highly guarded secrets. It only makes sense that Shiro would want to keep it hidden, as secrecy would've been drilled into him at a young age. Perhaps also, Shiro wanted the focus to be on the art of judo and not the old world from which he came. We'll never truly know Shiro's intentions, but the evidence is overwhelming.