If the Sharingan can see chakra transmission and channeling, then how come Kakashi couldn't see or deduce that Pain was being controlled as a Puppet when they fought?

So I've seen/heard people in Uchiha hypothetical Vs battles [when it comes to Pain] assert the notion that a Sharingan user would be able to deduce or see the chakra being funneled into the Pain corpses[through their receivers that look like piercings], and either figure it's a minimal gain to defeat them all in battle, or even ascertain Nagato's whereabouts. I assume they mean there would be a faint visible trail to trace into the sky or wherever Nagato would be hiding. I also assume that this notion is derived from Itachi in the fight against Sasuke when he stated the Sharingan can assess a person's chakra levels and see their chakra. Or, from Sasuke when he told Deidara the Sharingan differentiates chakras in the form of colors, etc. But if that were true, Kakashi would have been able to accomplish this. Despite him having a borrowed Sharingan, he could do everything[within reason] a regular Sharingan user that inherited it and awakened it through possessing Uchiha blood could do, he even managed to achieve MS, and copy 1,000 jutsu (according to lore). As intelligent as Kakashi was, if he could do this, knowing he couldn't take on all the Pains and win, he would've simply disengaged from the fight altogether, and attempted to meet up with reinforcements to storm Nagato's hideout and take him down. Which leads me to the answer of, the Rinnegan puppetry is apparently unique, in that it cannot be visually traced, and does not function like the basic Puppet Master jutsu that Kankuro and Sasori utilize where there is an obvious visible set source of chakra in the form of strings being used to control the medium.