In most gaming scenarios the difference between an i3-3xxx and i3-4xxx is pretty minimal. We can look up benchmarks of the i3-3xxx CPU's and see they always perform at least as well as that A10-6800K when factoring in a dedicated GPU--
Tom's has the hookup. The fact is, i3's are better for gaming, despite not being focused on gaming because their per thread performance is so much better than AMD's APU's. The only time an APU is a better deal is if you're using the iGPU.
TechReport backs this up.
...and the opposite case can be made. The AMD offerings are fully unlocked, and can be pushed significantly more. You cite two sources, but CPU Boss comes to the opposite conclusion here:
http://cpuboss.com/cpus/Intel-Core-i3-3225-vs-AMD-A10-6800K. My point is simple, the AMD offerings are superior for gaming because they are unlocked and focus on higher clocks. Yes, Intel is more power efficient. Yes, Intel has better IPC. Yes, Intel has significantly better fabrication processes. All of these reasons are why Intel can charge a premium for the i5 and i7 processors. I buy them whenever I get the option, because the high end AMD offerings aren't up to par.
None of Intel's benefits are currently competing well against AMD in the low end market. Lesser IPC, boost the frequency. Poorer lifetime, the socket lasts longer than 36 months. Stock settings not competing with Intel, then use that unlocked clock. I've seen plenty of stock comparisons where AMD has its silicon handed to it. I've also seen overclocks that make up for every shortcoming and then some. If Intel allowed an unlocked clock, then the i3 would be a contender. As they aren't unlocked, there isn't any reason that I can see to justify them in a cheap gaming rig. Gaming rigs are meant to push boundaries, not run at stock settings. If you're looking for a stock gaming rig, then please run an i3.
I can't, in good faith, recommend an i3 for a gaming rig, it's that simple. If I were building my own, I'd get the 6800k. If I were less pressed for a budget, there'd be a Haswell based CPU in my system. There's no reason to be a fan of one company over another, and that is the only reason I'd see in favoring a stock i3 (because they aren't unlocked) to an AMD processor that can be overclocked.