Hello,with the auger belt off grab the auger pulley and check to see if it wobbles indicating a bad bearing.
the pulley doesn't wobble or feel loose everything seems tight.
Re install the belt and check belt/pulley alignment.Sounds like something is not lined up right or bent causing belt to come off.Pull the rope when belt is on and handle squeezed to check.Look at idler arm and pulley too.
just put another new belt on it. cleaned all the pulleys and checked alignment pulley seems to wobble at speed
doesn't seem to be wobbling enough to cause the belt to come off
All springs are on and belt has tension?
Run it with belt cover off and auger engaged and try to see where the belt comes off at.If it is the pulley that wobbles a little replace it.If there's enough play in the idler arm it could case the belt to come off it and get chewed up.