If you don't want to be sliding all over the road in ice & snow...that's not a fault of AWD. You describe a no traction issue which falls directly to the type of tires you have mounted. A 2WD vehicle mounted with dedicated winter tires will outperform an AWD vehicle mounted with all seasons or summer tires.
If you have all seasons or summer tires mounted...an AWD system is not going to be able to do what it was designed to do...if you don't have traction...you're not going to fare well...and again, that is the work of the tires.
Second, your AWD system has a permanent 38/62 torque split (front/rear)...and it uses DSC to help. If your ABS/DSC system isn't working properly...this is adding to your issues. Unlike the facelift e53s which got the variable torque split iDrive system...the early e53s got an AWD system that was similar to the old AWD systems used in the e30/e34 "ix" models...but with the addition of DSC. Sometimes you will see it referred to as
DSC-X...but it was short lived since iDrive debuted around the 2003/2004 model years with the e90/e53/e83.
Below is some info about the early AWD systems that used DSC-X...perhaps it will help explain some of your driving issues that go along with probably the incorrect tires mounted for the driving conditions: