Ugh, I always forget I can't just upload images to this forum. So here are the benchmarks quickly:
Unigine Heaven Benchmark 4.0 - Extreme preset, plus the now ancient X-Bench Quartz, Open GL, and UI Tests:
NVIDIA GeForce GT 710 - Zotak (Mojave):
FPS: 4
Score: 102
Min FPS: 2.5
Max FPS: 8.9
X-Bench overall UI score: 32.99
RX560 - Sapphire (Mojave):
FPS: 22.4
Score: 565
Min FPS: 6.9
Max FPS: 48.6
X-Bench overall UI score: 30.66
RX570 - Sapphire (Mojave):
FPS: 42 (what else could it be?)
Score: 1058
Min FPS: 8.2
Max FPS: 94.8
X-Bench overall UI score: 34.43
RX580 - Sapphire (Mojave):
FPS: 43.1
Score: 1086
Min FPS: 8.3
Max FPS: 101.4
X-Bench overall UI score: 56.72
I also decided to run the old X-Bench set of 3 UI benchmarks since I didn't quickly find anything else to perform a similar test. When I was using them I was unable to perceive any difference whatsoever, and only the 580 actually showed much difference in the benchmarks (being quite a lot faster). Interestingly the 710 outran the 560 on the UI tests by a tiny margin.
Problem Points:
The 710 does NOT support audio over HDMI, while all three of the AMD cards did.
All of the AMD cards detect our monitors incorrectly and set the output to YCbCr instead of RGB when hooked up with the old DVI to HDMI cables we previously used. They work fine HDMI to HDMI and things look great over DisplayPort too. This wouldn't be a problem because we *should* be able to simply switch color modes like Windows users have been able to easily do forever, but of course we can't do that in the Mac world without jumping through some unbelievable hoops. 
This may be a problem for some users. If you use multiple-monitors be sure to get a card with enough other outputs where you won't need to use the DVI output of the card. The 710 did not have this problem.
I tested all four of these cards in a MacPro 4,1 flashed to 5,1 and with the original dual-quad-core E5520 processors, and also booted the system into Catalina public beta using DosDude1's patcher. All 4 of these cards ran without any glitches or visual artifacts of any kind in both Mojave and Catalina public preview, which I think is a good sign.
So, at this point, it looks to me like any of these cards is a viable solution for every day use, with the main considerations against the 710 being no HDMI audio and absolute uselessness for gaming. The RX560 also performed poorly in the Heaven's "Extreme" setting, but did perform OK on the basic setting, but I forgot to record the results and I'm not going to bother doing so now. Suffice it to say, if you want any chance of quality gaming you have to go with the 570 or 580, but the 560 might suit older titles OK. All of them performed poorly enough at the absolutely-critical "Min FPS" test that, realistically, you may have to make a lot of quality concessions regardless of which card you use.
