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It is easier to control kittens rather than hooman children.
His primary focus would be saving Rin and Obito, for emergence of Pain I think Zetsu and Madara played a big role for that, so they used just enough pressure to crack Nagato's psyche without getting him killed. They wouldn't have let Nagato be killed at the very least risk damaging Madara's rinnegan. The plot has already changed in big manner because of how talented Takeshi is.
They will experience Pain!
Hizashi is a Jonin from Hyuga clan and he is head of the Branch Family, he had a valid reason for blocking Takeshi's chakra point, so nothing would happen to him. But if he had done any permanent injury on Takeshi it would've been another thing altogether. Not to mention the mocking he would get from Uchiha clan, the Hyuga main family elders might just decide to sacrifice him to clean such a disgrace.
True he took the easy way out, but he is on a timer, with the timeline he is in, he will be sent off to war soon enough. Trying to recreate Rasengan using cat's chakra network would take him years atleast, if this wasn't a random black cat and rather Nekomata from Cat castle, then he would've had the chance to achieve that because for that cat humans war's wouldn't matter and he had more than chakra for trial and error. But then this would've become Cat Kingdom building story.
He would need to invent a method to be able to use it using his cat body, the chakra network of cat is different, not to mention rasengan is chakra intensive, people outside of elite chunin chakra level's can't use it. The rasengan he receives from system would also include experience on how to use it well. So he can decide how big and chakra intensive he wants it to be.