Does beating hot oil into eggs before cooking an omelette make a difference? Usually, when I prepare an omelette, I pour oil in a cast iron pan and heat it.
In the meantime, I prepare the eggs and I beat them fluffy. When the oil is pretty hot, I pour it in the eggs while beating them and finally I cook it in the pan.

As long as I can remember, I've done that following my mom's example. I assumed that it makes a difference but today my wife asked me why I was doing that.
I couldn't find an answer other than "I always do like that".

I'd like to know if this process have any value on the end result. If so, I could give her a better answer. If not, I will change my habits.
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!