It does not need to go to earth. Take a 1.5v alkaline battery, connect one end of the battery to the other end – a large amount of current flows, no earth involved. The electric charge that a neuron can produce is basically like tiny cells of a biochemical battery. The problem is unlike a useful battery, the voltage difference between all the individual cells is not (and realistically cannot be) carefully organized in a series or parallel path from positive to negative, instead all the positive and negative connections are jumbled together into a complex network, meaning there’s no way of getting billions of volts out of it. It’s just not wired that way.
Theoretically if you carefully constructed a series of hundreds of billions of neurons connected end-to-end-to-end in the right pattern you might end up with billions of volts (although end-to-end it would probably be the size of the solar system, so the billions of volts potential wouldn’t seem so impressive anymore on an astronomical scale) and you probably can’t pack it your neural-battery into a small space without the neuron’s insulator (myelin sheath) from breaking down and shorting out that voltage. Also it wouldn’t really be a brain anymore at that point. The complex maze of connections are what makes the thinking happen. If you make them all single-connected you’ve basically just got a really big, low capacity and relatively inefficient battery compared to better chemistries.
Flowing all that current at once will certainly create a lot of heat though, you’re right about that. That heat is normally heatsinked by the intracranial fluids and conducted away by the relatively rapid bloodflow through around the brain to be dissipated in the skin and lungs. The brain is basically liquid-cooled and it’s a very efficient and tightly regulated system that rarely has issues. Such a high neutral output would probably overwhelm even the relatively robust cooling that bloodflow provides, though, leading to a condition called brain hyperthermia, which is part of the reason drugs like methamphetamine can be dangerous or fatal, as it can result in cell death, and in this case, probably brain death and overall death.