There are 2 circuits for the coil, one to give it power while
your cranking, and another one to give it power while in the
ON position. One way to check this would be to see if the car
push starts. The other is to temporaritly run a wire from the
12v + terminal on the battery to the + on the coil directly.
See if it starts after that, if it does your cranking supply is suspect.