Why doesn't the extra C wire work when hooking up WiFi thermostat?
Check the voltage between the R wire and C wire at the thermostat.  I bet it's 0. 

It happens all the time... installers hook the wires backwards on the transformer.  Historically, it didn't matter because transformers make AC power. Polarity doesn't matter on AC.** As long as one side of the transformer went to "thermostat" and the other side went to "furnace", it was good.  The early transformers didn't even have R and C labels, that was added later as a "convention" for future expansion.  And the convention is "R" to thermostat, and "C" to furnace. 

Well, these smart thermostats need power for themselves.  Since they're supposed to be getting "R", the instructions say connect "C".  Of course if your transformer is wired backwards, the terminal labeled "C" on the transformer already goes to the thermostat, so connecting another wire does nothing.  

So on "C", what the smart thermostat is looking for is the side of the transformer which goes to the furnace (and not to the thermostat).  Whichever that may be.

** When I say polarity doesn't matter on AC, I mean for this little low-voltage stuff.  Different deal on 120/240VAC household power because they do certain tricks to improve safety.