You might be lucky. I don't think you will see anything unusual in the oil, if the engine oil cooler was leaking. But you would see oil sludge and/or oil scum around the insides of the coolant recovery tank.

Maybe a bad engine oil cooler isn't a part of your problem. But I'd be surprised if your 2011 Aveo has lasted all the way to 2021 with its original engine oil cooler. They always end up leaking sooner or later.

Deano