Yes, but you're missing the point, the only rinnegan ciruclating here is Madara'sFirst, we still don't know how exactly Nagato has gained his Rinnegan. Everything we know that Obito told that he have it to Nagato, but it's a lie. Until we have a proof, it's only a consideration about changing the name to Madara in this statement.
"Must have"? It's all your ossumption, not the fact.
Still no proof, though it's indeed most likely that his eyes were transplated, it's just logical, but we still don't know the sircumstances of this.
And sure the conclusion is not correct. We don't know when Madara's died, and even if you were right, transplating the eyes doesn't mean death. As we all know, Izuna died not because of his eyes lost, he continued to fight for some time.
Therefore Nagato's rinnegan is Madara's
Madara even said ''that brat Nagato finally managed to grow up''
And no, you miss the point again, I'm not saying a transplant means death, I'm saying Madara was reanimated in the state he was before his death with his eyes.
So where is the problem?