Think of the atmosphere as if it were an ocean. You might not think water has weight if you were diving underwater, but obviously when you fill up your cup with water you feel its weight increase. The atmosphere is really just a gaseous ocean on top of the surface. In extension, if you were to light a candle on the edge of a building taller than the Earth's atmosphere (assuming you had an oxygen source), you would see the smoke fall towards the Earth.
But these particles never pile up on each other and push a surface down by their weight so that we can measure it as weight, not pressure.
This is simply false. Gaseous particles do exert forces on surfaces. It's just that the scales we have are designed to subtract the cumulative forces of air particles. If you were to tare a scale in atmospheric pressure and then place it in vacuum, it would give a negative reading because the weight of gases are no longer present.