Well this is the silliest thing I ever heard of, not the prop that's pretty neat but the mom falling in the toilet. my wife and I have one son and between the two of us we have been reminded many times after we forget to put the seat down. I don't in all of my adult years remember my wife ever falling in the toilet, I guess she always looks first. Maybe someone needs to invent a "butt prop" (short pc. of 2x4 strapped across her backside) the mom could wear to prevent here from falling in.
I don't think it's about the fear of female falling with the seat up.
It may be presented so to underline the extreme.
The main, latent, reason is that us men lift the seat in fear of wetting it with our piss. Now, women are such extreme hygienists that they try to avoid touching the seat with their hands in the first place, hence the imminent dilemma.
I saw the same kind of solution by a South Korean Company, only there it's hydraulic with a touch of a button.