PS 3 is a really really old machine with extremely dated hardware.
It only has 256 MB of system RAM!
I'm sorry but no way is it going to be faster than a $105 to $110 Core 2 machine with E5440 quad core , 2GB RAM and GT 630.
Ya how about no?
I have a Core 2 Duo with 8GB of ram and a 9800M GTS and the graphics of games are not equivalent to the PS3/Xbox 360.
As for the thread title, my idea is APU/Integrated graphics. Low Profile/Cheap GPUs in todays market at the price bracket you're talking about, I'd rather just have stronger integrated graphics. I think Skylake and AMD strong AMD APUs come close.
The market for $50 graphics cards is so tiny, hence why neither company puts much effort into it.
Edit: Especially since you're talking about new games like BF4, there is no way that system will handle it. My Core2Duo system I played Dark Souls at 30 FPS and below, I played Skyrim at 720p and got low frame rates, etc.
You're better off with a console, but hey, you'll do what most people do who makes these threads, you'll do the exact opposite of what every person in the thread says to do and come back later asking for help/reasons why your system won't perform and blame everyone but yourself.