Just to double check, when you say you measured 24VAC, did you measure it from the common (COM) wire to that zone, or some other points? The correct way is to measure between common wire and that zone. I really can’t explain what you are observing: if the voltage between common and that zone wire is 0, either there is a shorting on the wires (or the solenoid is shorted), or the triac switch is damaged open (essentially leaving the circuit open). However, the fact that you can measure 24VAC without connecting the zone wire means the triac switch should be working.
There is a way to directly test the triac. Specifically, use the multimeter’s diode testing functionality to measure the diode forward drop voltage between gate and T1 pin (red probe on gate, and black probe on T1). You can find the pin layout on the datasheet:
http://www.ween-semi.com/documents/BT1308W-400D.pdf
page 3 (note that gate is marked 3 and T1 is marked 1). If the triac is ok, the forward drop voltage should be about 0.6 to 0.7V. Otherwise the triac is damaged.