The gear does look pretty rough but it is just worn, not broken. Start be seeing of there is any "wink" in either belt pulley. One should move until the belt stops it. the other may not move at all. If the motor seems to be moving, look towards the chain end. Work your way through it until you see what is stuck. Taking the belt off may be the way to start but since you want to replace the pulley anyway, going at it from that end may be better. That doesn't look like the 3 GDOs I have here to look at so I will be short on details about how the drive works.
I imagine there is a you tube these days telling you how to go at taking this apart.