I wouldn't say it's stupid, but it is loaded with potential downsides. A Xeon dual CPU ready motherboard may be expensive AND still not suitable for gaming. You have to invest in the power supply capable of powering it AND in two cooling solutions for the CPUs. Already, just from this stuff, the low cost PC might already lose whatever advantage it had in low cost. Finally, in the end it may not even be what determines game performance depending on the games you will play and the videocard you buy.
My advice is don't go farther back than Sandy Bridge. Try for at least a dual core w/hyperthreading and a decent if older videocard, you'll be happier. It can cost less if you get a pre-built, like a Dell Optiplex 790, and just add a videocard. if you buy it piece by piece and aren't patient looking for deals, it can wind up costing more although you do get greater flexibility to choose what you want.