Get the FE 2060 and run it with adaptive sync. Freesync is open source, the only step required is to have Nvidia implement the feature in their GPU drivers, which the are trying(kinda) but they don't want to loose the extra moolah coming from G-Sync monitors.
As for the Rx580, the FE2060 will run circles around the rx 580.
The adaptive sync is kinda the middle ground when it comes to syncing. Its not as bad as V-Sync and not as good as the freesync.
The Rx580 cannot go beyond 60 - 80 FPS in modern games, so why bother.