Mining rig with 12x560s would mine with the same speed as 6x570s, but you will use more space and double of motherboard/CPU/RAM/HDD/rigs/etc.
Maybe going with Nvidia is a decent option these days when no AMDs can be found.
RX 560 is just half the rx 570. Half the price, half mining performance, half power consumption. I believe 2 of this cards are equal to rx 570.
Yes you need to take 12 cards to make hashrate like 6x570/580, but when you can't fiend cards this is much better alternative then nvidia. Besides when the new asrock mobo with 13 slots arive you can save from cpu,ram and ssd.