Not following how the Baal hatanya being learned all over should make me think tzaddikim are infallible
Not all tzaddikim, a tzadik gomur of the Tanya is extremely rare!
Nearly all tzaddikim are either a tzadik sheaino gomur or a beinoni of the Tanya, both of which are very lofty levels, but are certainly not infallable.
For reference, Rabbah thought he was a beinoni, as defined in the Tanya.