I know of no particular problems with the mid to late 2000s Ecotec.
It's a solid engine that I'd own myself, if only it came installed in a car I'd actually want to have to drive.
I would imagine that most of the owners of these car aren't especially flush, which is why they bought them. Many of these cars that supposedly need engines may need only simple repairs.
It may be that Cousin Vern gave the current owner the bad news and Cousin Vern may know as much about diagnosing an igniton or fuel pump problem as my mother does.
So, lacking the bucks to have a competent tech diagnose the problem or to have a good yard engine installed if that's what's actually needed and needing to pay for something else to get to work, the hapless owner puts her (these are kind-of girl cars) Cobalt on CL hoping to get a few dollars out of it.
You'll also see a number of "needs engine" Hondas on CL because the hapless owner failed to have the timing belt replaced before it failed, along with a number of "needs engine" BMWs, because the hapless owner ignored the temperature gauge and cooked the engine after a minor coolant leak.
It ain't just abused Cobalts on CL.
It's every make and model of car you could imagine, along with a number of good machines of every make and model you could imagine.
In short, the Ecotec is no worse than most engines and is better than many.