Further on that - there's just a few differences between an ABS electro-hydraulic module and an ABS+TC electro-hydraulic module. Since your ABS works fine, that helps rule some things out.***
ABS-only: 8 solenoids
ABS+TC (for pass cars with diagonal split braking systems): 12 solenoids - that's 2 extra solenoids per circuit to allow the unit to build/control brake pressure above the level of the master cylinder.
Normally a fault in one of these, such that it would disable the ASR system, would also generate a fault code. ABS is preserved as long as the 8 ABS solenoids are not affected.
Communication with the engine - The newer the system, the higher chance that it also modulates engine torque as well as applying Left or Right brakes when the wheels spin. Older systems have a dedicated signal wire pair between the ABS/TC module and the engine controller (usually a PWM signal) and newer systems communicate via the CAN bus. Not sure what a '95 A8 uses here... but generally this kind of failure should also set a fault code.
Interface with the TC Lamp. Sometimes hardwired, sometimes via CAN bus. Obviously your light is working
Interface with the TC switch. Sometimes hardwired, sometimes via CAN bus. Given lack of diagnostic codes, I'm wondering if this is a more likley culprit. Most of the systems that I worked on did have diagnostics that would detect a open, shorted, or stuck switch, however maybe a '95 era system does not? Given reliability of mid 90's Audi electrics, might be something to look into.
*** one test I would do, if easy enough, is to find a long stretch of snow or ice, and do a higher speed (if safe!!) 80kph snow or 40kph ice ABS stop all the way down to zero. If your pedal gradually falls towards the floor as the stop progresses - this might indicate wear to the pump/motor portion of the hydraulic unit, rather than focusing you on an electrical problem. (though that said, sometiems the connection between the motor and the electrical controller is external and could be subject to corrosion, if damanged, and the motor isn't getting proper current, could lead to the problem above , and the TC might be smart enough to detect that and disable itself -
In fault modes, ABS in general is left on as long as it can be for safety reasons, even if performance might be less than 100%, where TC is generaly disabled pretty easily with a slight fault.
Sorry this isn't very specific to Audi, but hopefully might help you troubleshoot.