If you absolutely cant find the schematics (After searching long and hard) it may be best to rewire the gocart for an acceptable controller ie one that will handle the voltage and current for the cart. This may mean losing some functionality and a little creativity in terms modifying an existing controller but its not that hard and it could be done in way that can be redone if you find the original controller later. This would comprise of identifying all of the control modules on the cart and their respective wires and matching their functions to those of the new controller. The idea being to start with the most obvious and work your way back to the least obvious. For example the power in from the batteries and the power out to the motor should be fairly obvious. This takes care of 4 wires to the controller. then try your best to match up the remaining wires to their counterpart on the cart. This is what I would do though not everyone would feel comfortable with this approach. At least I hope it gives you another option in case the original controller proves impossible to find. Good luck
Have you tried measuring the battery pack Voltage with the power on and throttle fully engaged to see if it drops down? Usually when a controller clicks and the motor does not run it is from the battery pack Voltage dropping under the controller's low Voltage cutoff (LVC) level.
A bad brake switch is another possible cause of the controller clicking and the motor not running. Unplugging the brake lever switch from the controller will emulate a good brake switch on most go-karts. However, some go-karts may have a normally closed brake switch so if unplugging the brake switch from the controller does not help then unplugging the brake switch and jumping the two terminals together in the controller's brake switch connector is the next step to take. If neither having the controller's brake connector terminals open or closed helps then the brake switch is not likely to be the problem.
The throttle can be tested to confirm if it works or not using the guide on this page: https://electricscooterparts.com/throttletestingguide.html
The motor can be wired to the battery pack to confirm that it works.
If the brake switch and throttle are good, the motor is good, the controller is receiving power from the battery pack, the power switch is good, and the battery pack Voltage is not dropping under 23 Volts with the power on and the throttle fully engaged then chances are pretty good that the controller is faulty.