As far as handling goes, I've made negative comments on how my Aveo originally handled in the snow, and to some extent on wet pavement also. Once I put good snow tires on, that problem went away. And I actually have always liked the way it handled on dry pavement - not a sports car to be sure, but I also didn't pay a sports car price either .
And on the timing belt, you need to know the Aveo is no different than any other interference engine that has a belt instead of a chain. A belt breaking in any vehicle so equipped is very likely to trash the engine and cost mega bucks. I used to have bad dreams (not kidding) about the belt breaking in my daughter's Honda Civic. I changed the belt, wp, etc, and the bad dreams stopped. Yes, there was some premature breakage on the 2004's, but I'm not aware of any systemic problem with the subsequent year Aveos. Hope you will begin liking your Aveo more as time goes on