I would check to make sure that the black ground wire coming from the charger port is plugged into the brass colored terminal on the back of the power switch, if the black wire is plugged into the power switch and the switch does not light up when it is on then the light has burned out inside of the switch. The switch could be replaced with a new one in that case. However if the light inside of the switch is burned out that would have absolutely no effect on whether the Crazy Cart runs or not.
Since the controller is clicking that indicates that the fuse and power switch are both good, and since the battery pack is new we can presume that it is good also.
Regarding getting the Crazy Cart running, from the parts that you have replaced and your description of what is happening when the throttle is pressed it sounds most likely that the throttle has failed. It is a common problem with some Crazy Cart versions for the magnet inside of the foot throttle to come loose and prevent the throttle from working. When this happens the controller will click but the cart will not run when the throttle is pressed.
Some people on this forum have been able to glue the magnet back in place to get the throttle working again however I have just read about this and have never done it myself so I can not offer any specific advice on how to get to the magnet or glue it back in place. If your other Crazy Cart has the same foot throttle then you might want to consider hooking it up to the cart that is not running to make sure that the throttle is actually the problem before attempting to repair or replace the throttle.
Please let us know how it goes.
Since the battery pack and controller are brand new, and the throttle has been tested, and the controller is clicking when the throttle is pressed which indicates that the power switch and fuse are good, the only part left to look at is the motor.
Like the magnet in the throttle coming loose, another common problem with Crazy Carts is that their motor wires will break from the steering wheel being repeatedly turned back and forth while the cart is being driven. Usually only one of the wires will break and the break will be under the plastic wire insulation so it will not be noticeable by looking at the wires. On some newer versions of the Crazy Cart there is a metal spring over part of the motor wires and the wires can break under this spring.
The first thing I would do is turn the power switch on and press on the foot throttle, then slowly turn steering wheel to its full left and then full right position. Sometimes the motor will power on at a certain position that the steering wheel is in, and other times it will not. If the motor powers on during this test then that definitely proves that one of the motor wires is broken. If the motor does not get power when the wheel is turned back and forth that neither proves that the motor's wire is broken or is not broken, and a further inspection of the motor wires are required.
To inspect the motor wires use your hands and bend the individual motor wires back and forth in one inch increments all the way from the motor case to the plastic connector at the end of the wires. The section of the wire that is broken will have less bend resistance than the rest of the wire does. There should be black plastic sheathing over the wires which will need to be removed in order to test the wires. If there is a spring over the wires then it will need to be removed or pushed out of the way so all of the wire between the motor and connector can be tested. Once the bad section of wire has been discovered then it can be spliced back together with an automotive wiring crimp connector and electrical tape can be wrapped around the wire to keep it from bending so much in the future.
Here is a photo of a Razor Crazy Cart with broken motor wire.
Hopefully this information will help you to diagnose the problem and fix the cart. Please let us know how it goes.