Questions about tweeters in 2009 Aveo5 LT

So I bought my Aveo in 2013 and wanted to upgrade all the audio but never really got around to it. I finally decided to get a new receiver for christmas and put it in today. I got it from crutchfield and told them I have the 6 speaker version with tweeters. On the wiring harness, though, there was only one pair of wires for front speakers, and I wasn't sure if that's enough for the door speakers and tweeters). And after I hooked it up, I swear there's no sound coming out of the tweeters. Just for the hell of it, I decided to unhook the new one and hook up the stock unit again. Still nothing from the tweeters! Go figure.

So for one, any idea what's up? I mean I bought the car used, and all the speakers were already in bad shape (I need to replace them all). Could they just be completely blown and nonfunctional? The electronics weren't in perfect shape when I got the car. The clock also resets and the tire pressure sensor doesn't work. The stock head unit takes around 10 tries to get it to accept a CD. So I wouldn't be incredibly surprised if they didn't work at all.

If they are completely busted, and I have to replace them, do I need a harness with wires for the tweeters? Here's the one I got:

Metra 70-2105 Receiver Wiring Harness Connect a new car stereo in select 2007-11 Chevrolet and Pontiac vehicles at Crutchfield.com

I'm new to car audio, so thanks for the help. For now, I'm just psyched to have a working clock and CD player, but I'd like to get the speakers sounding nice eventually too.