The log shows that the Epic Games Launcher is trying to query the available builds from Epic’s servers, but the request fails with an HTTP 504 Gateway Timeout error.
This means the server is not responding in time, so the Launcher interprets that there are no applications associated with your account , which leaves the install button locked.
I checked several forums and found the following:
Some users reported that the issue was fixed by opening the Launcher from the taskbar with right click then Library (as shown in the video I shared above).
It is also recommended to temporarily disable antivirus or firewall software, as they may interfere.
Others mentioned that the problem solved itself after a while, since it appears to be related to Epic’s servers.( Now it seems that the servers are working correctly, so I don’t think this is the problem.. )
If none of this works, the best option is to contact Epic Games Support so they can check the issue on their side.