In a way, theres two answers. Chronologically, the first one is because he couldn't, or wasn't allowed, and afterwards, there would be no need to.
From the Naruto Wiki on Rasen-Shuriken:
After travelling a certain distance or colliding with a target that it can't slice through, the central sphere of the Rasenshuriken detonates, producing a vortex of wind in the immediate area. Within this vortex, countless microscopic wind blades are created, so many that even the Sharingan can't count them all.The wind blades pierce every cell in the body, disconnecting them from the Chakra circulatory system and so preventing them from providing their physical energy to the chakra-creation process. Because of this, the target loses their ability to create and mould chakra, damage which cannot be healed by medical ninjutsu.Tsunade likens the damage to poison, and warns that if the damage is great enough, those affected by it won't be able to use jutsu again. (emphasis mine)
Because the jutsu damaged the user itself, Tsunade marked it as forbidden, which is why he wouldn't be able to do that, as I previously mentioned. It was forbidden. So, he wouldn't be able to do this because the jutsu was marked forbidden. Most jutsu, such as the Shadow Clone Technique, are made forbidden because they can severely damage or kill the user. Rasen-Shuriken was an example of that. So of course, he wouldn't do it with a clone. Naruto didn't even have permission to use it.
And as to why there would later be no need:
Naruto is able to overcome these flaws by learning senjutsu. By infusing the Rasenshuriken with senjutsu chakra, its shape is maintained after formation, meaning that he can use it as a projectile and, as a result, that he no longer needs to worry about damaging himself. (emphasis mine)
So, one he couldn't do that because it was marked forbidden, and later, when he mastered Sage Mode, he wouldn't need to because he himself can throw and maintain its shape upon entering it.