[JohnnieB]

The reed switch and the battery charger not turning on are two entirely separate issues.
They can get confused because they both use the third wire in the plug or receptacle, however the third wire in the receptacle (reed switch circuit) connects to the positive side of the battery pack while the third wire in the charger's plug (battery sensing circuit) connects to the negative side of the battery pack.

1. Reed switch: I suspect the reed switch was bypassed by a previous owner because the reed switch failed. If the solenoid clicks when you push the pedal, it was a successful bypass.

If there is a white wire, or any other color wire connected anywhere within the battery pack other than one of the two main terminals, it is probably for 12V accessories. (Which is a bad idea, a separate 12V or 36V to 12V converter should be used)

Here is a drawing showing the main battery pack terminals label B+ and B-

2. Battery charging: The heavy white wire from the charge receptacle connects to the main positive terminal (B+ on drawing), it may or may not have a 50A or 60A in-line fuse in it. The heavy black wire connects to the main negative terminal (B-).

If you measure the voltage between the two female terminals in the cart's charge receptacle, it should read exactly the same as the voltage between the main + and the main - terminals on the battery pack.

For the charger to turn on, it must sense the battery pack voltage.
The second attachment is a schematic for a Powerewise 28115 charger with the battery sense circuit highlighted.

The positive side of the battery pack (Red highlighted) is sensed via the wire that connects to the positive source inside the charger.
The negative side of the battery pack is sensed via a wire that is separate from the wire that connects to the negative source inside the charger (highlighted in Blue)

The negative side is sensed by the Auxiliary contact in the charge plug touching the negative terminal in the cart's charge receptacle.

Your charger turns on when you wiggle the plug in the receptacle, so the most likely suspect is the Auxiliary contact in the charger's plug, which most likely has not been replaced, cleaned or adjusted.