This is the way! It helps me to imagine what would it look like if the atmosphere consisted of a single nitrogen molecule. You place it on the ground but the ground has temperature (is warm) so your one molecule gets launched up into the vacuum on a parabolic trajectory at 500 m/s on average. If it launched at 45° it would reach 6km up and fall down, at 90° - 12km up - and that's on average. Some would get launched faster and higher (following the long tail of the Boltzmann distribution), and hydrogen and helium even faster still because they are lighter. A few hydrogen molecules would be launched at speed above 11km/s, which is above Earth's escape velocity, so they would escape and never fall down.

When you have many air molecules, they hit each other on the way up (and down), but because their collisions must be perfectly elastic, mathematically it works out that the overall velocities are preserved. So when your one nitrogen molecule gets launched up but on its way hits another identical molecule, you can think of them equivalently as passing through each other without colliding at all. (Yes, mathematically they can also scatter in some other random directions, but the important part is that your original molecule is equally likely to be boosted further upwards as opposed to impeded.)

The end result is that majority of the atmosphere stays below 12km, density goes down as you go up though never quite reaching zero, and hydrogen and helium continuously escape to space to the point none are left.