I've got an '09 5-door (hatchback) and I've never gotten stuck in snow (except for one time, briefly, on an un-plowed street with snow over a foot deep), although the original tires are getting worn down now at about 18K - because they are crappy tires. Original tires that come with most new cars are bottom line in quality, never optimal.
The Aveo has a really primitive suspension - it's a cheaply made car. That's why rear rubber suspension bushings wear out and cause annoying problems. Don't expect it to handle like a Corvette, or even a Cadillac. But an Aveo still can be a fun car to drive, despite the cheap design and construction of it.
However, if you put high quality tires on it, it should grip the road better. When it comes to tires on any vehicle, you get what you pay for.
As the old saying goes: "Always buy the best. You'll never be disappointed."