Does a water heater have to be completely full for the water to heat?
Any leak large enough to stop the water from filling the heater tank would have to be so large as to basically be an open ended line. It would be impossible not to notice the flooding caused by a leak this large. A smaller leak would not stop the heater from filling and operating and you would have some hot/warm water delivery. I am not saying you don't have a leak, but I strongly suspect an electrical issue. It could be related to power supply, or it could be related to damage caused by improper heater installation or operation. Call a reputable professional plumber please.