The CPU doesn't hold up at all. For gaming, AM3+ wasn't really recommended even in 2015. The AM3+ CPUs woefully underperformed and in gaming workloads, i3s and then the least expensive Ryzens were regularly beating the most powerful, overclocked AM3+ CPUs.
To be fair, the same could be said for the competing i3s and i5s that were in a similar price range back in 2015. In late 2015, it should have been possible to get an FX-8350 for less than the cost of an i5-6400, and closer to the cost of an i3-6320 when on sale. Those locked i3s and i5s may have been a bit better for gaming at the time, but the FX-8350's roughly 33% higher multi-core clock rates, even at stock, prevented those parts from pulling too much ahead, unlike Intel's more expensive unlocked parts.
And since then, core counts have been increasing, and a lot of the more recent AAA titles really want more threads than those older mid-range Intel processors have to offer. In a game heavily utilizing four threads, that i5 might still manage to get over 20% higher frame rates provided it's paired with a higher-end graphics card, but the i3-would be choking with its mere two cores with SMT, preventing it from being much better than the FX-processor, and probably worse in terms of frame stability. And in a game like Warzone, that's been shown to run best on processors with more than 4 threads, even the quad-core i5 would likely choke a bit. There are plenty of modern games where an i5-6400 will still outperform an FX-8350, though an increasing number are likely to see benefit from the FX procesor's higher thread count, even if ultimately neither processor is particularly ideal in 2020. And with a GTX 970, many modern games will be limited more by graphics performance than anything, so you generally won't see the kinds of differences between processors shown in reviews utilizing much higher-end graphics cards.
It's probably worth asking, have you tried running your existing hardware in Warzone and other games to see whether performance might be suitable enough? And have you tried lowering graphics settings somewhat to see whether that might lessen demand on the graphics card and improve performance to any significant degree? This hardware isn't exactly ideal for gaming at this point, but should still run the vast majority of games fairly well, provided you are willing to turn down settings a bit and are okay with dips below 60fps.