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Also having him check, but wouldn't that kill the 3rd brake light as well?
No. They two are usually on separate fuses so if you blow one fuse you still have some sort of brake lights. I cannot read the schematic provided by Dadaroo to determine if hazard lights working means the fuse for the main brake lights is good.
8VETTE7 you got me thinking (potentially dangerous) about my 2011 Silverado as I thought I remembered it having multiple fuses and I've encountered other cars with two fuses. So I was forced to GOYAGL (Get Off Your Azz And Go Look)
My Silverado actually has one fuse for each brake light. Evidently it wasn't the thing to do in 2003. At least not on the Corvette. (verified that too)
I think I found the cause. The customer reported he previously had the euro-type tail lights on his car. If this is the case, he simply needs to re-connect the white wire on the brake switch.