If you look at the two wiring diagrams for this circuit, you will see there is only a fuse for the washer pump. The wipers have a relay but the washer does not. I traced the power flow in red. You should have an open circuit (0 volts) until the switch is closed. However, if you notice this is after the load so your pink wire should have minimal voltage on it even when the switch is closed because the wiper motor uses the voltage and all that is left is ground. Due to the switch being on the ground side, you are correct that you should not have constant voltage on the LG wire. The only why I see that happening is if the washer motor is shorted to ground, the switch is shorted (or stuck on), or the wire prior to the switch is shorted to ground.
I suspect this is what burned up the pump as well because even though you don't have the switch on, the pump has power so it should be running.
Here is the easiest way I know to test the ground wire and switch. Put your red volt meter lead on the Pink wire at the pump and then put the black lead on a known good ground (either the ground for the washer pump or the negative battery post) and then do not press the switch and see how many ohms you have. It should be open or no continuity. Then press the switch and close the circuit and you should then have a small amount of resistance, like less that 0.5 ohms. If this is what you have then your wiring and switch are good and it has good ground. This means it is your pump that is shorted.
Images (Click to enlarge)
Friday, June 7th, 2019 AT 6:19 PM