First off, it's not like the motherboard vendors are going to deliberately try to deceive you by saying a CPU doesn't work with a motherboard when it actually would. They don't care whether they sell a board with a newer or an older chipset; if it's in stock, they'll sell whatever they can.
Second, while it may not be immediately clear that they've kept that page updated, with a little digging I can tell you that they have kept it updated:
-- The motherboard uses AMD's 880G chipset. That chipset's first release was 2nd Quarter 2010; the SB850 southbridge this particular motherboard uses had its specifications released by AMD in December 2010 (http://support.amd.com/TechDocs/44758.pdf). So you're talking about a 6- or 7-year-old model here.
-- Most of the supported FX CPUs listed are the early Bulldozer-based CPUs, & per the site require version F3 of the BIOS or later. That BIOS, per the site, was released October 20, 2011. Note that this closely matches the release dates for those early FX chips (FX-4100/6100/8100 released by AMD October 12, 2010).
-- The newest CPU on that supported list, the FX-4350, was released by AMD on April 30, 2013. This was well after the FX-6350, 8320, & 8350 were available to consumers (all released the same day, October 23, 2012, that the supported FX-4300 & 6300 were released). Since they already had other 6- and 8-core FX CPUs supported on the board, there's no reason for them to have omitted them without any testing...which means they probably did test them & found they wouldn't work.
Because of that, I could maybe see the FX-8320E or 8370E CPUs working, mainly because they're 95W TDP chips. But I wouldn't swear to it, & I wouldn't be surprised if they didn't.
In the end, based on the list of officially supported CPUs (http://www.gigabyte.in/Motherboard/GA-880GMA-USB3-rev-31#support-cpu), I would recommend the Phenom II X4 980 over even the FX-4350; the FX might be slightly faster, but it's only a 2C/4T CPU, while the Phenom is a 4C/4T. In some cases, I might even go for the Phenom II X6 1100T; slightly slower but a 6C/6T CPU. Unless you really, really need some capability in the FX-4350 (or even possibly the FX-6300), that motherboard is old enough that the Phenoms are probably better.