You seem to miss a core point. One thing is when something is a weapon at your arsenal, one other thing is that something is an external factor that does help you in one war or another, but it's not within your entire control.
IN the Narutoverse, several shinobi use the body, or body-parts, or genes, of another shinobi as weapon. As long as they can do it, yes, it is "their" power. Having a living weapon that is FORCED to do your bidding no matter what they think, is blatantly different from having an ally.
Sasori uses the Third Kazekage, Kakuzu uses hearts from other shinobis, Pain uses summons and corpses that move like if they were part of him, Danzo uses organs and genes stolen from the Uchiha and the Senju. Are those "their" powers? Of course they are.
Is this different from Madara controlling the Nine-Tails, or Obito controlling the jinchurikis, like if they were their own limbs, parts of themselves? Of course it's not. Sure, both Obito and Madara can be separated from these human weapons, and therefore lose that power. But this doesn't mean that they aren't theirs as long as they possess and control it, just like any other weapon.
Edo Tensei is a different subject. Edo Tensei clearly doesn't grant you full control of every person you summon. It has been demonstrated several times already that highly powerful/strong willed shinobi can break free of it like Hanzo, Sasori, Itachi, Madara, Hashirama, Tobirama all did. Indeed this is why Kabuto left some of the reanimated shinobi free of moving, like he did with Deidara, Ginkaku, Kinkaku and some others.
Edo Tensei doesn't necessarily give you access to the full abilities of the reanimated one... see when Kabuto decides to un-seal Deidara, because he "doesn't quite get his art" = he wouldn't know how to fight with Deidara's style even if he had Deidara under his full control.
On the other hand, if you want "full control" with Edo Tensei, the seal possibly ends up weakening your own reanimated warrior, like when Orochimaru first resurrected Hashirama and Tobirama against Hiruzen Sarutobi.
Obito was the only one who demonstrated a full control of his reanimated shinobis and their abilities, but he was able to do that because he had combined Edo Tensei with sharingan, rinnengan and the Gedo Mazo's ability to control the tailed beasts. And even in that case, the effort to control all six jinchurikis made him lose the ability to use other powerful techniques such as all the Six Paths techniques.
In short...
Generally speaking when someone is entirely in control of another person's body, you should count the powers of the controlled as powers that the controller has access to. In Edo Tensei's case, though, it's not exactly that simple.
