Does beating hot oil into eggs before cooking an omelette make a difference?
Not tried or heard of this before. I can only arm-chair experiment in my head until I try this out. This seems to be an encapsulation/spherification method.

I can see a thin "shell" of cooked eggs coagulating around the hot oil droplets, trapping them and preventing separation. With many tiny beads of egg encased oil (effective oil buoyancy reduced) in suspension and thus more fats locked in the eggs there is likely a richer mouth-feel. This would not be achievable otherwise. Pretty neat trick. Your mom sounds like a modernist cook!