I would not buy the RX580. To increase a little the clock, you have to feed it quite a bit more power.
RX480 can indeed already beat the 1060 in some games.
Anything that includes Rise of the Tomb Raider is useless, as this game seems not optimized at all for AMD.
And 3GiB VRAM is not enough for some games at 1080p Ultra.
Right, as I said the 480 can beat the 1060 in a handful of games, but overall the 1060 wins more than it loses. That's why I said it's better on average. It's not really fair to exclude a game just because it runs poorly on AMD, using that logic I could just exclude any game that runs better on AMD and claim that NVIDIA wins 100% of the time. Granted, if all you care about is that one specific game, then by all means buy the GPU that runs it the best, but this is rarely the best course of action.