It sounds to me like the amp is overheating. I wouldn't be surprised if the gain on the amp is set to high and you are clipping the signal at loud volumes. Does it shut off randomly or when you are listening at loud levels for long periods?
A poor ground would not cause the amp to shut off but if you are interested in hitting high SPL numbers I would do the following. Replace all the power and ground wires with 4ga. Make sure the amp and cap are grounded to the chassis. I.E. do not daisy chain the ground wire for the amp to the cap first.
For your ground point, make sure to sand down to bare metal, use a locking washer, bolt it down real tight, and cover the exposed metal with silcone to prevent rusting.
Checking the remote turn on wire isn't a bad idea either. If the amp is turning off randomly that could easily be the cause.