You would see a lot of benefit by raising your budget by $100 or so, but it isn't nearly as bad as some these guys are saying...
PCPartPicker part list / Price breakdown by merchant
| Type | Item | Price |
|---|---|---|
| CPU | Intel Pentium G4400 3.3GHz Dual-Core Processor | $49.99 |
| Motherboard | Gigabyte GA-H110M-A Micro ATX LGA1151 Motherboard | $49.99 @ B&H |
| Memory | Crucial 8GB (1 x 8GB) DDR4-2133 Memory | $27.99 @ B&H |
| Storage | Toshiba P300 1TB 3.5" 7200RPM Internal Hard Drive | $39.99 |
| Video Card | Zotac GeForce GTX 950 2GB Video Card | $119.99 @ Micro Center |
| Case | Xion XON-310_BK MicroATX Mid Tower Case | $22.99 @ Amazon |
| Power Supply | Corsair 430W 80+ Bronze Certified ATX Power Supply | $22.99 |
| Operating System | Microsoft Windows 10 Home OEM 64-bit | $84.99 @ NCIX US |
| Prices include shipping, taxes, rebates, and discounts | ||
| Total | $418.92 | |
| Generated by PCPartPicker 2016-06-13 00:38 EDT-0400 |
All the manually entered prices are from Micro Center. There's an additional $10 rebate on the mobo. If you go with this, check their inventory online to make sure they have it all in stock (there's 2 Micro Centers in Atlanta so I wasn't sure which one). I did break $400, so if that's a problem, sub in a GTX 750 ti or R7 360 to save $20. The 950 is a good buy at that price, though. The dual core is a problem in some games, but is more than fine in most, including all the ones you mentioned. You can easily upgrade later on if you want.
GTA V - with worse GPU at 1440p: https://www.youtube.com/watch?v=BxEucCBENHQ
Doom: https://www.youtube.com/watch?v=2zhbvY6-GIE
Fallout 4 - with worse GPU at 900p: https://www.youtube.com/watch?v=QYBfl9li_ak
Hard to find examples with this exact setup but this should give you an idea of what to expect. Very respectable performance, I'd say.
Also, before anyone goes crazy over the CX PSU, they're really not bad. The CX line has been through several revisions and the last one put in a Panasonic main capacitor for Pete's sake! Anyone that's ever recommended an EVGA 500B shouldn't be hard on CX at all. Ripple is absolutely stellar, and voltage is too at medium load. This will go nowhere near max (and even at max it's good); we have over 200w of headroom.