Not a surprise it doesn't work. It's a bogus formula. It assumes the tires have no effect at all on the load-carrying capacity and the tire footprint. They have a very large effect on both.
The tire pressures listed in your manual are those which will allow for uniform wear on the tires at a certain load, plus keep them safe. These have been skewed in recent years since increasing the tire pressure increases the MPG of the vehicle by reducing rolling friction. Since manufacturers like to advertise as high an MPG (and avoid gas-guzzler taxes in the USA) - they tend to set the tire pressure higher than optimal for ride/handling/comfort/wear.
It's a whole bunch more complex than simply dividing PSI by weight.. although I guess that sounds like a nice idea.