Welcome to the forum. It would be helpful to provide the specific model etc for your generator, but on my 4kw Onan gen the "status" +12v lamp/LED line is connected to the generator control board. I don't have a schematic for the control board but someone on the forum may have it or could provide it.
So either the control board has an internal failure/short which only affects the status lamps/LEDs on remote and local gen start switches, or the status line has a short to some 12v source external to the control board.....which does not affect the control board apparently becuase you say the gen starts and runs normally. I would bet on bad control board.....easiest to change (but probably not cheap).
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