Zwitterions/IEP of glycine at pH 6? (Paradox?)
You need to separate the two concepts.

When dissolving a neutral amino acid in water, a buffered acid-base reaction will take place as you highlighted, giving a certain pH value. In this case, the following equilibrium:

$$\ce{HOOC-CH2-NH3+ &lt;=&gt; ^{-}OOC-CH2-NH3+ &lt;=&gt; ^{-}OOC-CH2-NH2}$$

will be somewhere between the pure second and the pure third species. (The neutral, non-zwitterionic species is neglectable in aquaeous solution.) Superfluous protons liberated into the solution will end up as $\ce{H3O+}$ explaining the overall reduction in pH value when compared to a neutral solution. You won’t notice it unless you perform really accurate experiments, but the solution’s pH value will be a tad higher than the isoelectric point.

If you go on and add external protons, i.e. by carefully adding an acid, you can adjust the overall solution’s pH. Since the glycinide anions are the strongest base present, they will take up the protons first, followed by the zwitterions. At a certain pH value (corresponding to a certain external proton concentration added), the concentrations of glycinide anions and glycinium cations will be identical and we have reached the isoelectric point of glycine. It is, and has to be, slightly different from the pH value obtained by dissolving glycine.