Why is it of high importance to control the center of the board in Chess?
In addition to the points you made, controlling the center means that your opponent will then have to form his or her attack on the wings.  Think of not only what it does for your pieces but what it means for your opponent's.

Picture the movement pattern of a knight - if you control the center, the knight can attack all eight squares.  If your opponent's knights are forced along the side of the board they're likely going to be in the first, second, seventh or eight columns (files).  That means they're less effective. ("A knight on the rim is grim.")

If you have control of the center, it's usually the time to start planning an attack.  Has your opponent castled?  If so, start getting your pieces aligned to make an attack on the appropriate side.  If you're playing more of a positional game it's time to start marching forward, taking up real estate on the board and placing your pieces and pawns onto squares that make it difficult for your opponent to move/develop his or her pieces.