4 Wires in Switch Box - Ceiling Fan/Light
The non-live black braided wire is not neutral.  Wires are not neutral simply because a voltage tester doesn't light up.  The non-live black is onward always-hot.  It became de-energized because you disconnected it from supply when you separated it. I trust you did not do that hot :) 

Take the blue wire out of service. Cap it off at both ends. Tape the cap down (they fall off single wires otherwise).  That's the only time you ever tape a wirenut. 

The LINE/SUPPLY black from the new switch joins the 2 cloth switches.  

The LOAD/TO-FAN black from the new switch joins the one unattached wire (red). 

Up in the fan, connect the fan controller to red and neutral.