Watch the Right-Hand Wrist online guitar lesson by Bruce Arnold from Guitar Physiology Survival Guide
The wrist is not used to pick the strings. The wrist should float freely and should not rest on the bridge unless you are doing a special technique like muting of the strings. As mentioned this is the most common way people play the guitar but it's also the most common reason people stop playing the guitar or are severely limited in their improvement. Some people can play their whole life from their wrist and never have a problem, while others end up with severe pain which can blow up to a point that they can't even open a door. First problem with playing from your wrist is the motion of going side to side with your wrist is probably the worst thing you could do. This is the exact movement that cause carpal tunnel syndrome. Another problem is your pick accesses each string at a different angle unless you move your wrist to the same angle for each string which most people don't. This makes for uneven sound from string to string because your wrist is changing the angle of the pick depending on which string you are playing. While this can be compensated for by changing the angle that the pick hits the string if you play from your forearm you will find that you don't need to make this compensation and will be one of the first things you notice in your playing. It will sound more consistent and smooth. You also want to be careful not to tighten the wrist as you play. Many players tighten their wrist to gain temporary speed. But this can't be maintained and usually ends in catastrophic failure of the wrist which can take years to heal.