Why doesn't atmospheric pressure affect weight?
Weight is the force we experience pushing up from the surface by our mass and gravity. So, our weight in Newtons is our mass in kilograms multiplied by the acceleration due to gravity
Air pressure is all around us. Yes, it presses down increasing our weight. It also presses up on our upward facing surfaces which cancels it out.
There is a small difference in weight due to air pressure, but most of it is cancelled out.