I'm not sure what your expectation is for the traction control (TC) and ABS systems. If you're thinking it's going to be like AWD, you'll probably be disappointed. In general, TC limits the slip of the drive wheels (front on the Corolla) during take-off and acceleration by electronically limiting the engine power/torque to those wheels. It does not provide traction that was never there due to ice or other slippery conditions. Similarly, ABS will prevent individual wheels from locking up due to lack of traction during braking but cannot perform magic if the conditions are such that very little traction/friction is available (such as on ice). The ABS will start to engage as soon as one or more wheels start to lock up. If you happen to be on ice or other very slippery surface, all the wheels may be slipping and the ABS will have the effect of locking/unlocking each wheel repeatedly which will make it seem like all the wheels are locked and sliding (which they are for all intents and purposes). As drivers in very cold snowy climates will tell you, the only real solution for ice is studs or chains but these solutions can make dry road driving more dangerous (or impossible in the case of some chains). At the very least, winter-specific tires are ideal for these conditions, but not so great on pavement which is dry or just wet.
One major factor in all of this is your tires. Modern all-season tires are, at best, a compromise and they vary widely in quality and performance (especially in snow/ice). Unfortunately, Toyota has a history of putting relatively low cost/low performance all-season tires on Corollas. So, if you have a tire with poor wet/snow/ice grip to start with, it will have the effect of reducing the performance of the TC and ABS dramatically. In fact, with very poor tires, the TC and ABS will both engage much sooner and more suddenly than they would with a better tire.
The OE tires on our daughter's 2009 Corolla (with the TC/ABS option) were so poor for both wet and dry traction that we replaced them almost immediately when the car was new with higher quality/performance all-season tires.
Hope all of this helps explain what may be going on. As I said above, a new set of good all-season tires might solve most of your concerns if they're in your budget.