Lone,
Unless the Human Interface in the system does as it is supposed to do by Procedure, Policy, Practice, and Experience....no system devised will be effective.
Case in Point is the Cougar Tragedy....had the Captain followed the Checklist Procedure (which the Co-Pilot advocated to do...) then there would have been a much different outcome to that sad affair.
To validate your point about False Indications....you probably recall the Blade BIM system on the Sikorsky Iron....which had both a Blade Indicator and an In-Cockpit Warning Light. The Blade Indicator was prone to give false indications occasionally....but the Cockpit Warning was notoriously prone to lie. My outfit removed the Cockpit system and went with a visual inspection of the Blade BIM's every time we shut down and set a Time Limit between Blade Checks.
There will always be False Fault Indications I would guess...but a False Positive is an inconvenience or nuisance whereas a False Negative (No Warning) could be quite deadly.
As to your SH-2 Crew....being on the ground at their maintenance base....shutting down and calling for a Tech or even a Tug would have prevented that Fatality.
Did not the USMC not lose a HueyCobra at Twenty Nine Palms not so long ago from a similar decision to continue flight following some kind of fault warning rather than land out and call for technical assistance?