It probably doesn't say that Le Chatelier shifts the equilibrium because it's more of a rule of thumb than an actual explanation of why things happen. You could figure out the direction of the shift using it, but using the Kc helps you to figure out the numeral change, while Le Chatelier just points you to the right direction.
Think of it this way; at the beginning, you have 2 moles of NO2 and 1 mole of N2O4. When the pressure is changed, let's say you end up with 1,8 moles of NO2 and 1,1 moles of N2O4. If the change in pressure was brought about with a change in volume, say from 2 liters to 1, at first the concentration of NO2 would be 1 and after the pressure increase it would be 1,8; while with N2O4 it would first be 0,5 and then 1,1.
If you think of it mathematically, since we often use numbers like 0,2 or 0,05 as concentrations, squaring a concentration actually makes the number smaller, so the [NO2] has to increase more than [N204] just to keep Kc constant.
This didn't answer your question fully, but I hope it helps at least a bit!