Glad you got it running again with no issues!
If you still have the old part you'll see that tab is just an extension of the inner tensioner body. Although it does bear against the outer tensioner body, that's only before the tensioner is installed into the belt loop. Prior to installation it serves as the "stop" which prevents the spring from unwinding itself. But after the tensioner is installed into the belt loop and the WP is turned to tension the belt, this tab permanently moves away from the stop position. It essentially becomes a free-standing piece of metal just attached to the inner tensioner body, and doesn't bear on, or make contact with anything else. I can't imagine how this tab could break off without something else happening first, but also wouldn't expect a tension loss to occur if it did break off. However, if it did somehow break off with the timing cover in place, then the loose, pinballing tab would probably find a bad location fairly quick and destroy something else in the belt loop.
I'm not saying all of this to nit-pick what you wrote, or argue about what happened to your Aveo. It's just that there have been a number of "mystery" timing belt failures posted here, and the waters are usually murky regarding what really happened to cause the problem. There was another failure just last week which was described as a tensioner "losing tension". But when I asked further, there was no reply confirmation that the tensioner spring broke.
What's really going on with these belt problems is a concern to those of us who DIY this job, because if there's a defective part on the market, it would be great to identify it so that everyone could stay away from it. But OTOH, if it's something going wrong with the installations, that would be a big help to those who might avoid doing something incorrectly.
When you opened up the timing cover for the first time, did you happen to attempt to turn the WP before loosening the 3 bolts? I'm asking because I believe the easiest installation mistake to make is leave one or more of those bolts loose, which then leads to the WP moving, tension being lost, and the belt jumping. So if you could move the pump without touching the bolts, that would point to a mistake on the install. Just trying to get more information about this chronic problem.