The problem I see is that the HD 5770 is not supported with AMD driver updates since it is considered to be Legacy status while the RX580 still is supported by AMD Drivers.
In my opinion, you can't have one Legacy AMD Driver and the latest AMD Driver installed at the same time unless the Legacy driver is compatible with the RX580.
Here is the last newest legacy driver for the HD 5770. BETA Crimson 2016 which isn't compatible and doesn't support the RX 580: https://www.amd.com/en/support/kb/release-notes/rn-rad-win-non-gcn-16-2-1-beta
Here is the latest RX 580 driver which isn't compatible nor supports the HD 5770: https://www.amd.com/en/support/kb/release-notes/rn-rad-win-21-6-1
