If you had a bad battery, there would be lots of other symptoms, like chugging under a load, inability to operate at all, or something. You definitely need to put a meter on the battery pack and make sure that it is somewhat normal (37.9 or so without a load and not dropping more than a volt or so when moving the cart), but if that is ok, and you still have problems with that contact overheating, there has to be something wrong right there. Does the motor spin the wheels when on the jack stands? If, for some reason the motor won’t spin, that would change everything. A stalled DC motor with voltage applied will draw excessive current and cause all kinds of things to overheat. The cart kind of internally hemorrhages trying to get that motor to spin.