So you want a gaming PC for $350 because that is what is left after OS and Monitor.
Here is the bottom line, it is fully possible to create a $350 computer that can play games on low settings. The problem is that this budget platform tops out at less then medium performance and you have to spend a bunch more money to get out of it. Thus if you ever want to upgrade it, it will take another $350 to get it at the level of a $500 computer.
So by getting just another $200 of budget you can actually get a decent platform to start with, that would then last you through several upgrades