You really need a multimeter for any serious troubleshooting. A test light isn't accurate enough.
Your problem could be caused by a loose or corroded connection at the battery, a defective or blown power wire fuse, or a loose ground wire connection. I'd probably start by checking the battery connection, and replace the fuse on the power line near the battery, even if it looks good.
A capacitor won't fix a power problem, especially in a case where the amplifier isn't even turning on.
Your problem could be caused by a loose or corroded connection at the battery, a defective or blown power wire fuse, or a loose ground wire connection. I'd probably start by checking the battery connection, and replace the fuse on the power line near the battery, even if it looks good.
A capacitor won't fix a power problem, especially in a case where the amplifier isn't even turning on.