Sigh... Yep, and It could be sooo simple like the glow plug light on old Mercedes diesels. When the light goes off, the oil pressure is at, say 70%. When you start the engine and the pressure builds up from the mech pump, the aux pump shuts off. You could have six figure odometers like the old MB oil burners too.
Of course, and when the pressure falls below a certain value, it would come back on. That way, hot oil, hot day, idling engine in traffic would have the advantage of keeping bearings flooded and lifters pumped up.
It would definitely increase engine longevity.