I found the problem - at least on my bike. Faulty side-stand cut-off switch.
After stripping the bike down and checking every connection, I started thinking about the things that are
supposed to prevent the engine running under certain circumstances. The infamous side-stand switch came to mind immediately. I unplugged it, the fuel pump spooled right up, and the engine started instantly.
So - Any of you G5/C5 riders who experience this syndrome when you turn the key: lights come on, neutral light comes on, starter turns over fine, plenty of power from the battery, but no engine light and no sound from the fuel pump - just unplug the side-stand cut-off switch and see if that fixes it.
After taking a break while checking connections, I came back out to the shop and found that, as usual, if you start work on a motorcycle, spectators show up out of nowhere: