My money is on faulty controller at this point, but I think you'll need to find someone with electronics chops if you want to avoid just buying parts until it works again.
For what it's worth, I didn't mean take the sensor out of the wall, but just electrically unplug it from the controller to see what it does on its own when you turn on the water.
A frequency counter won't really help you here, I think. You already know to expect ~VCC/2 when water is running, and either VCC or 0V if it isn't. The speed of the square wave isn't very relevant.
5V or 4.68V input isn't meaningful. The sensor has some input range and 4.68V most definitely falls into that. Could be a design choice that has no real implications.
On the other hand, if the device normally supplies 5V, just yours doesn't, then that's further evidence you have a faulty controller.