The problem following the board indicates the problem is on the board, but I'm not sure both boards are bad.
I glanced through the TC III manual and it made no mention of the delay, so the control board used inthe TC III and TC IV may act differently than most do and coming on immediately may be normal.
A Radsok is a type of high current, low insertion force connector. The male halves are in the charger plug and the female portions are in the cart receptacle.
EZGO and their charger suppliers got a bit tricky and used the outside of the negative female connector to sense the battery pack voltage, via a spring like third contact in the plug.
Here is a picture showing the receptacle with the top off. The female Radsoks are the brass thingies.
Note: The positive one (white wire) typically has an insulating sleeve around it.
Also, here is an exploded view of the plug. The auxiliary contact is item 36.
The control in the charger that has ran for 11 hours and still putting out 13A, might be good.
If at all possible, try it on known good batteries before buying a control board.
If fact, try both of them, bad batteries make charges do weird things.
