Found Windows 10 Fix for Black Screen

Open a Windows Explorer window and navigate to the place League is installed ("C:/RiotGames" is the default path) and look for "LeagueClient.exe"

Right Click on it and select "Properties", then select the "Compatibility" tab on the window that pops up

Check "Compatibility mode" in that menu, and ensure the Windows XP (Service Pack 3) option is selected.

No amount of firewall, reinstalls, driver updates etc did anything. But this? Boom, all issues resolved.