That's indeed an interesting perspective; I just double-checked in sane units:

Standard pressure 101 325 N/m^2 divided by standard gravity 9.80665 N/kg gives the mass of the air per area: 10 329 kg/m^2.

Divide that by an air density of 1.225 kg/m^3 to get a height of 8435 meters[1]. The Mount Everest is 8848 meters high. If the atmosphere were uniformly dense, the highest mountains would rise above it.

([1] Not in fact accurate to four digits because the constants I used aren't really constant across that height range.)