Battery testing tools are normally capable of showing if a battery is good or bad. However, as I already said above, there are some uncommon situations where a cell is bad, and the tool can't pick that up. I would lean in that direction when the battery is losing static voltage, and load testing shows bad voltage recovery numbers, as your Aveo is doing.
Doing another battery load test is definitely a good idea. And if the results are the same as the first test, that would reinforce the theory that something is wrong with the charging system. I'm saying that (if it were mine) I would buy another battery in this situation based on the fact that the current one is of unknown age. The logic I'm using is that the battery will need to be replaced sometime fairly soon anyway, so it won't be a waste of $$ in the long run, even if it doesn't fix any of the current issues (assuming the intent is to keep the vehicle).
With a new battery installed, I'd be looking for the load test to show good, steady voltages, and that would confirm the existing battery is bad. But if the load test results don't change, then the problem in the charging system is somewhere else.