Receiver

Separates DO work better! Its just that you have to go very, very far up the food chain for that to be the case. Where money and value are concerned, that is where integrateds really shine. 

The main difference is that with a stereo integrated amp you can at least achieve higher performance with separates as long as you are willing to spend a lot more money. Where with multichannel there is no amount of money that will buy you as good quality sound as you can get from something as relatively affordable as say a $3k Raven Blackhawk.  

With a setup like the OP has, and assuming a very limited budget, the smart thing to do is first answer the all-important question: What matters most to me, really? And I mean REALLY???! 

Because if it really is sound quality then the answer is stereo and the value answer is an integrated amp. But if it is "Surround! I must have surround! And 7 channels! I want HOME THEATER!!!" Then in that case forget about it. Might as well stick with the AVR. Do Whatever. Just don't kid yourself about sound quality being a priority. Its not. Its for movies. Its a visual medium. And everyone making components for HT knows this, they know their market, and they pay just enough lip service to dupe buyers into falling for what they all want to believe, that they can have it all. 

If you care about sound quality then you need to know the lowest component at the very bottom of the audio food chain is the AVR. Next up from that is the stereo receiver. All these things cram way too much stuff in way too small a box and with way too cheap parts to be work well at all. Take your money, which obviously if you are even considering one of these things you worked hard for, and buy a stereo integrated amp. Vast improvement. 

Or forget sound quality and get the AVR with the most features that looks the prettiest.