I am sure the RX 580 would be fine on an i7 870 machine as it's a mid range card. If there was any bottlenecking it would be minor and not noticeable.
The whole bottlenecking thing is massively overblown to the point of being absurd with the mid range cards and below.
With the higher end cards like the GTX 1070 and above, yeah some of the older CPUs would be a problem pushing them.
Here are some information/scores from my two machines in 3Dmark and 3Dmark 2011 on the RX 480.
Used 3DMark because it's a controlled test, same every time and it separates the GPU score from the rest.
CPU's:
I5 750 @ 2.8 GHz
XEON X3450 @ 2.8 GHz
I7 870 @ 3.2 GHz
I7 7700K @ 4.8 GHz.
Graphics card - XFX RS RX 480 8GB at stock speed.
Scores are Graphics Scores only to highlight the GPU only:
3Dmark Fire Strike:
I5 750 - 13,528
XEON X3450 - 13,707
I7 870 - 13,517
I7 7700K - 13,922
3Dmark Time Spy:
I5 750 - 4,138
XEON X3450 - 4,139
I7 870 - 4,132
I7 7700K - 4,155
3DMark 2011:
I5 750 - 19,504
XEON X3450 - 19,409
I7 870 - 19,631
I7 7700K - 19,898
So much for an I7 870 bottlenecking an RX 480, I still say with the mid range cards any i7 (8 Thread) will be fine as the CPU isn't the bottleneck.
Now results could be different with different system setups, graphics cards etc.
With cards above mid range as in the GTX 1070 and above, yes I will agree with the bottlenecking issues.