YEP.
As said it depends on the game but my rough guess is you'd gain between 30% and 60% higher AVERAGE FPS with a top-end CPU depending on the game.
(It's hard to compare since more MODERN games tend to have CPU bottleneck tests with a higher-end card)
SKYRIM is more likely to have a CPU bottleneck (not well threaded) so here's an FX-8350 (close enough) and it's way behind an i7 of that year which is similar to an R5-1400 Ryzen CPU:
https://techreport.com/review/23750/amd-fx-8350-processor-reviewed/5
So the i7 at top of chart is about 70% faster with same GPU and settings... but in a more threaded game like Battlefield 3 or Batman AK there's less difference.
That's AVERAGE FPS. There are almost more stutters on average with weaker CPU's (called 0.1% and 1% lows in frame time tests).
RX-570 is about 80% faster so I'd say you'll gain at least 40% boost on average some higher some lower.
The higher VRAM (video memory) may be more important for smooth gameplay though I'd still hold off for a card with more than 4GB depending on how much you value the time vs money.
So... the RX-570 4GB should give:
1) higher FPS average for most games
2) less stutters if VRAM is an issue (i.e. 2GB wasn't enough)
3) CPU stutters that existed before would remain
SHOOTERS for example often rely heavily on the CPU.