Why is the trimethylammonium group a stronger electron withdrawing group than the nitro group?

I was working on a question that basically boils down to deciding whether the nitro group ($\ce{-NO2}$) or the trimethylammonium group ($\ce{-N+(CH3)3}$) on a benzene ring, is a stronger electron withdrawing group (EWG). 


The solution says that the trimethylammonium group is a stronger EWG since it has an overall positive charge, while the nitro group is overall neutral.


But I am getting confused here. Why should the nitro group be neutral? The $\ce{N}$ atom has 4 bonds (a single bond to an oxygen atom, a double bond to another oxygen atom, and another bond to the carbon of the ring), so the $\ce{N}$ atom in $\ce{NO2}$ should also have a positive charge? 


Further I analysed that since $\ce{O}$ is more electronegative than $\ce{N}$, it tries to shift the $\pi$ electrons of the double bond towards itself, hence increasing the electrophilicity of the nitrogen of $\ce{NO2}$, while in the trimethyl amino group, the $\ce{N}$ being the more electronegative atom gets electrons donated to it due to the +I effect of the three $\ce{CH3}$ groups, thus reducing the electrophilicity of $\ce{N}$. So $\ce{-NO2}$ should be a stronger EWG than $\ce{-N+(CH3)3}$.


Where am I going wrong?


