The AMD setup will definitely be worth it. the 965 competes with the i5 in performance. There are wins and losses. I believe the i5 comes out on top in the end by like 5% overall. For gaming there is no difference.
As for quality, AMD has been around for a long time and is reputable company. The 965 will last. Best for you to get the C3 (125W) stepping version over the C2 (140W) though. Less heat and power consumption.
As for motherboard quality, I trust ATI integrated graphics and currently, they are higher performing than the intel integrated (for IGPs) while at the same time producing less heat than the typical Nvidia integrated.
As for your "slowdown", a processor does not slow down with age provided you keep the case airflow good and clean/prevent dust buildup. Another cause of slowdown is the actual OS having been installed on the computer for a long time. A third is the Hard disk drive slowing down over time and eventually failing. If you would be happy using the same software and OS you had when you got your old computer and are willing to have the same performance as when you got your old computer, some simple maintenance and a new hard drive will get your computer back up to speed.
That being said, it's probably time for a new system anyway since your computer is 7 years old.
Another note: The motherboard you chose is a crossfire board without integrated graphics. Unless you plan to crossfire 2 graphics cards, I wouldn't recommend it. I use the AM+, non-EVO version of this board right now:
http://www.newegg.com/Product/Product.aspx?Item=N82E16813131406
So if you aren't planning crossfire, I'd go with that.