Reading your description, I'm quite sure that the problem is not the sensor circuit.
At 99% the problem is a broken mosfet; if the speedo needs to turn on that mosfet at startup the startup fails.
If you turn the motor slightly, then the speedo will start the motor using a different mosfet (different phase) and the motor start running.
Once that the motor is running it will continue running also if a mosfet is not working correctly.
Replacing a mosfet is a big problem in these kind of ESCs and cannot be done without some sophisticated soldering equipment (and also with that will be quite difficult).
Horizon will not offer a repair service because the xcelorin is actual some rebranded china stuff and the repair service will cost more than buying a new speedo
.
Bye
Ema
P.S. Unfortunately I got a speedo like yours with the same problem
.