The concept is pretty simple, but doing it takes a bit of precision and patience. Not sure if I understand you correctly, but you basically want to ONLY connect all 12v lines together. So you'll need to splice out pins 2 and 3 from side A, splice out pins 1, 2, and 3 from side B, and join the 5 wires together, then solder all of this to the one power cable. You need to be careful not to cut or break the presence pin (pin 1 on side A) since that is the one lone wire sticking out after the splicing.
ehrm, if you only connect the +12V what is going to ground it?
Hence no power delivered as expected, grounding would still happen via mobo and same burning effect will happen if that was required.
Grounding has to happen via some route. Probably the stupidest and most popular misconception/myth about electricity is "Grounding doesn't matter so mcuh, so it's ok if we use this 1mm2 ground with the 16mm2 power cable" (I've actually seen that with cars, barely any grounding, huge positive cable, end result: Burning wires)