Not really. It's ok if you want something you can carry around easily, take a few knocks, and support a full length graphics card, but "best ITX gaming rig" it comes up a bit short at that budget.
Also, your build won't work because the PSU won't fit in the case (well, it's 20mm beyond the manufacturer listed dimensions, it might still fit). Oh and some of your components are.
For me, something like this is going to utterly destroy it as a gaming machine. It's a tiny bit over budget due to component prices bouncing around a bit, but you can pretty easily rectify this with a few tweaks (save a bit on motherboard, cooler, memory or PSU). At this budget level though, an extra $40 is pretty insignificant when it enables you to include (arguably) the best motherboard, CPU cooler and graphics card money can buy.