I am not an expert so correct me if I am wrong, but I thought that the original power house of the Shias was North Africa (under the Fatimids of Egypt and Libya). It was not until the Safavids came to power in Persia in the 16th Century that Shia Islam began to be institutionalized there. So how could Shia Islam be an off-shoot of those Eastern faiths when it did not develop in those Eastern areas anyway?
Shi'ism's roots are mainly in modern-day Iraq (which in turn accounts for a large chunk of the old Sassanid empire). They were a persecuted minority in the region for a very long time, but the loss of Abbasid power (and, finally, the Mongol invasions) would make it possible for Shi'ism to flourish. The Zaidi Buwayhid dynasty that took Baghdad from the Abbasids (while nominally pledging allegiance to the Caliph) is a good pre-Mongol example. Post-Mongol invasion, local Shia dynasties would form in Iran - the Ilkhanids didn't feel the need to stomp on Shi'is - but it wasn't until Shah Ismail made conversion to (twelver) Shi'ism compulsory on pain of death after conquering Iran that Iran had anything resembling a Shia majority.
In short - Shi'ism never left the area, but it wasn't the officially mandated religion of a powerful ruler thereabouts until the Safavids turned up.
edit: Corrected ambiguous sentence.