TC: Please feel free to savage this theory?
I'll preface my augmentation of your theory by noting that in my opinion there is no necessity to shut down a failed engine if you are on approach in a helicopter (or in a light twin on short final for that matter) unless it is surging or actually on fire. An engine might surge while running out of fuel or when trying to digest a bad bit of fuel but it could also be caused by FOD (a bird strike?) or a compressor blade failure. If an engine did surge and the pilot responded by shutting down one, it's possible that he shut down the good one, inadvertently and tragically forging a necessary link in the chain of events.