No, I refer to Shiro Zetsu. Obito and Madara have created over 100000 beings with Hashiramas cells. By your logic nearly everyone would have Ashuras chakra, right?
Indora or Ashura doesn't influence the potential of the host either. Ashura and Indora chose only those with great potential to begin with. That's why Hagoromo said to Naruto: I can see why Ashura chose you!
Kuro Zetsu said-which is Kaguyas will-, Sasuke has more potential than his brother. Despite having disadvantages for his development.
Even if Itachi would have been a transmigrant it wouldn't change it. However, potential is also an individual propertie. Every circle of Ashura and Indora is different. At first Ashura was the 'good for nothing' guy and Indora was perfect. Hashirama was a great Shinobi since birth and Madara was inferior to him. However, Indora influences one important attribute~ that's the Choku Tomoe.
" Anyway to conclude, this can perfectly explain why an Edo of Madara and Hashirama can have both their prime Powers. With or without Ashuras spirit, Hashirama has the same chakra as his forefather "ashura" and so does Madara."
As I said before and have showed with that scan, Ashuras chakra exists alongside Naruto and isn't part of his own chakra concept.
Hashirama, Madara, Sasuke and Naruto since childhood kneaded their own inner chakra with physical and spiritual energy to transform it into the shape of ninjutsu. Their own inner chakra doesn't have the properties of Indoras and Ashuras chakra.
Your statement would make everything Hagromo said about the transmigration redundant.
If Madara and Sasuke would use a chakra exactly like Indoras to build Susanoo, it would have also the same color. That's a given fact. Since Susanoo consists completely of the users chakra. Same chakra= same color. Makes kinda sense. However, Madaras Susanoo looks like this:
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and Sasukes like this:
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It's different, because both of them use their own inner chakra which have different properties.
Madara and Hashirama are not strong because Ashura or Indora chose them, but Ashura and Indora reincarnated inside them because they had a great potential since birth. However, hopefully I didn't misinterpret your argumentation. :3