Air Bubba wrote - "I have seen both views argued in ETOPS training over the years. Of course, I've also been told that you have to proceed to your ETOPS alternate if you lose an engine even if BIKF is CAVOK and nearby (I don't buy this one)"
You are correct - alternates, whether destination or ETOPS, are for flight planning purposes only. There may be any one of several good reasons to not use a planned ETOPS alternate, that had been designated OK at point of dispatch or, of course, prior to ETOPS entry point, and go somewhere else. Hey, it may now be closed because someone else just did a wheels up a few minutes ago! As PIC you have both duty AND authority, including appropriate use of emergency authority, certainly under the US FAR's, to make the best decisions to protect the safety of aircraft and all aboard.
BTW, I fly the ETOPS 763, and I'm pretty sure the HMG is run from the centre system, in the event of AC power failure. To not have that back up "essential power" available, powering my basic flight instruments etc, would make me not want to wander too far at all from friendly surfaces, certainly within the (hopefully) 30 minutes battery life of aircraft.