Is this 05 C6 totally stock or is it possible in the past it was tuned as if so the tuner could have turned off certain DTCs ?
If stock, is it possible the DTC reader you're using does not support all GM enhanced DTCs ?
With key on, engine off, does the check engine light come on and stay on ?
At the least with 1 coil pack unplugged should cause misfires and that would trip DTCs for that
Really need is a full OBDII scanner and look the engines' feedback sensors and the fuel injectors pulse width times
along with the long and short term fuel trims and see if they report within normal range
Using a fuel pressure gauge connected to the fuel rail, (on end of left fuel rail) should report 58 PSI
Also assure the MAF and throttlebody are clean and no leaks from aircleaner to TB
Keep in mind the early C6s using the LS2 also had its own PCM model, which was not used once LS3 was used so
maybe the code reader you're using doing not support that PCM.