I'm not gonna read all the responses right now, but we don't yet know why the elder son got a slight mutation of the Rinnegan. Really, there are only 2 possibilities,
1. He got the Rinnegan 100% and the Rinnegan does not require senju DNA to be unlocked... BUT as soon as it was passed to him, it began to degrade and thus looks more like a spiral than the classic Rinnegan.
or
2. He only got the majority of what can be considered the Rinnegan and the full eye can only be unlocked but combining Senju DNA with it.
Considering it is possible to unlock Rinnegan at all, I think its safe to say that the FULL eye does belong to the Uchiha and does not require Senju DNA to unlock it, but we don't know for sure yet.
Seems to me that the best possibility is that the Uchiha did get the Rinnegan but as soon as it was passed down, it began to degrade and eventually resulted in the sharingan.