I've been thinking about this for a while, and I believe that Orochimaru most likely wanted an Uchiha body so he could get the Rinnegan to master all jutsu, but if he needed EMS wouldn't this have hindered him? Since he wanted to take Sasuke's body before he even got Mangekyo.
And this would explain why Nagato couldn't switch back to EMS from with Madara's eyes, because it isn't a simple evolution of EMS, it is a whole other dojutsu.
Madara also spoke like there was a possibility of Obito awakening the Rinnegan. He told him "Even if you don't awaken the Rinnegan" even though Obito didn't have EMS.
Maybe you only need Senju+Uchiha DNA w/ another hidden requirement that would explain why Madara got it once his natural life ended (Near death experience maybe).