I plan to have a separate 5/8" copper ground rod just for the spa.

I hope you meant an "ADDITIONAL" ground rod.

You spa ground must still be connected (bonded?) to the main service ground.

The purpose of an "additional" ground is to provide a return path to ground
for some "stray" voltages. It helps ensure that there is no significant
voltage difference between, say, your grill and the frame of the spa.