There are more things in heaven and earth then is dreamt of in your philosophy
Sasori's puppet of the 3rd kazekage was able to perform jutsu and had chakra of its own because of the way he was converted into a puppet.
The original 6 paths were in fact already dead, and Nagato was channeling his chakra into them via the chakra rods.
Madara's 6 paths are impure world resurrects. More than likely, unlike some of the other resurrects Kabto performed, these were likely fully mind wiped other than access to whatever powers they had (making them more 'user friendly' per se). and as resurrected beings, they would have fully functional chakra networks, so the chakra rods wont be needed. now the sharingan, rinnegan, and whatever else madara did to whatever was left of thier minds is another matter entirely
come fight my brute here