If the speed controller still works and only its blue motor wire's insulation has begun to melts and deform then the controller can still be used. If the controller to motor connector has been overheated and deformed then it should be replaced. If the blue wire has any exposed copper then that area could be covered with electrical tape to protect it from short-circuiting.
Other than adding extra weight to the scooter the 15Ah batteries will not cause any extra current to flow through the controller wires and cause them to overheat as they did. When going uphill if the scooter's speed becomes very slow then it is best to let go of the throttle and push the scooter up the rest of the hill to avoid overheating the scooter's electronics.
If needed we carry replacement controllers for the EcoSmart scooter on our Razor EcoSmart Parts page.
That would only protect against a short-circuit and not repair any functionality problems.
We recommend insulating the wires with electrical tape individually instead of insulating them as a pair.
The connector in your photo is not used on the EcoSmart so nothing connects to it.
Here is a wiring diagram for the EcoSmart scooter.
The new battery pack needs to be wired as drawn in on the photo below.
You could extend the black wire by splicing in a new piece of wire to make it longer.
It goes to the controller. We have updated the photo above to better reflect this.
Electric scooter speed controllers are too complex to easily test. Testing the other components that are connected to the speed controller and using the process of elimination is the best method to determine if the speed controller is working or not. If all of the other components that are connected to the speed controller test good then the speed controller has to be the problem.
We would start by turning the power switch on and testing the Voltage at the speed controller to wiring harness connector. The thick red and thick black wires are the ones to test. If there is no Voltage at this connector then the fuse, power switch, or battery harness may have a problem. If there is Voltage there then check its level with the power switch off and with the power switch on. If the Voltage level drops under 23 Volts with the power switch on then the battery pack has a problem. The expected Voltage at this connector is around 27 Volts if the battery pack and charger are good.
If the previous test does not uncover any problems then we would unplug the motor from the controller and connect it directly to a battery or battery pack to make sure that it works.
If that does not uncover a problem then we would test the throttle using the attached throttle testing guide. When using this guide the throttle needs to be plugged into the controller during the test procedures.
If the test results for all of the parts including the throttle are good then that points towards the controller being the faulty part.
Please let us know how it goes.