Is an i3 acceptable; yes and no. The question is at best nebulous, but more accurately poorly framed.
Do you want to run a 1920x1080, 1280x720, or a 3840x2160 monitor? The Raspberry PI can technically produce consistent frame rates at 1080p, albeit for movies. As 1080p monitors become almost ubiquitous, at what point does that become the default assumed resolution?
Additionally, who would spend 4 times as much on the GPU as the CPU? It's like taking a human, scooping out half their brain, and replacing the extra space with a set of gills. Assuming you live near the water, you'd be good at surviving there. Once you leave that specific area a bigger brain would benefit you greatly. Gaming is one specific use, in its current form. As that form changes (increased use of more cores, changes in system usage, etc...) the very specific evolution of a GPU centric system may not be viable for long.
So, why is an i3 viable? Crappy software. I've seen more modern games use less than two threads, than I've seen CPU utilizing beasts. As a thought experiment, lets play Fallout 3 with a stock i5 2500k versus an i3 2120. Both share the same frequencies, but the i3 is only a dual core. There is zero difference in the game (in my experience at least), as trying to run Fallout 3 on more than one thread results in concentrated failure. Yes, Fallout 3 is old. My point still stands because Gamebryo still stands. Companies are working with tech that has only recently began to recognize multiple cores. Almost none of it is truly well threaded, and that which is can still generally run on hyper threaded cores. There is no reason that 90% of games shouldn't run as well on an i3 as an i5, assuming all other factors are equal.
The real answer I'd like to give here is that Intel is not targeting the i3 to gamers. They've locked the multiplier, so the more expensive i5 is the choice for gaming. This is a reason to choose AMD over Intel for budget rigs. An i3-3225 and A10-6800k have the same price right now. The A10 is a bit of a pig, but it will overclock and it provides more physical cores. As AMD travels toward the same production tech as Intel, I see no reason why the i3 should be chosen over what AMD has to offer. If you want a gaming rig, a little bit of power waste is an acceptable premium for significantly more cores at a much higher clock speed. If you just can't purchase an AMD product, then the i3 can function in a gaming rig with some huge caveats attached.
They are starting to come down. I have seen just the 6 core cpu's going for $450 lately.
Where? Please enlighten us!