While pushing on the rear wiper spray lever does engage the wiper, the sprayer has its own motor and is not related to the wiper moving. You have 2 problems.

Most often, the rear wiper slows down until the corrosion of the shaft becomes too great and the wiper gives up.

Without taking the rear door panel off, you can try to spray some PBBlaster on the shaft and try to initiate movement by hand after turning the wiper on. But that is all you can do.

The wiring goes next to the left hand hinge and does need to be inspected.

Listen for the washer motor when you trigger the fluid release. The pump is below the fluid bottle on the right front fender. If you turn the radio off you will be able to hear it.
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