He (The Last Prophet) was the offspring of an Imam, The Prophet Abraham to be precise whom The Qur'an testifies for. (Surely, We have made you an Imam of men). If you don't think the sacrifice of Imam Hussain was a pure sacrifice in the Quranic sense, then you can do away with the obvious parallel between 'the ransomed sacrifice' that the Quran talks about in regard To Ibrahim and Ismaeel.
Your whole hajj is based around the movements and events in the Life of Abraham, the Prophet of God, and The analogy to that in the Modern Prophetic sense is to Mohammed and his family, IF you care to take note of the 'Durood Shareef' within your own Salaat.
So, if you fail to ponder upon the obvious reality then your own concept of purity in the form of 'Islamic Personalities' are concerned is also 'flawed' and you have not fully grasped or put into perspective the time honoured teachings as per narrations from The Qur'an in relation to 'family' and 'prophet'.
There were lots and lots of generations between Prophet Ibrahim and Prophet pbuh, so by that standard then any Fatimi can have a claim of being Mahdi, doesn't have to be direct off spring of previous Imam.