You don't have the folder yet because the game doesn't get that far into generating it.
Tells us which of these 3 happens.
You press play. Launcher closes, no image appears on the desktop and there is no process in task manager
You press play. Launcher closes, no image appears on the desktop and there is a process in task manager for archeage.exe
You press play. Launcher closes, an image of a poem appears. It disappears before all the text can light up
So it does do of them. It does the first one.
This happens only when something is preventing the game from launcher or outright terminating the launcher process.
The things that cause it are not running the game as admin
Not setting anti virus exclusions correctly. (Post your exceptions list)
Fault within the installation. Give you've reinstalled multiple times we can rule that one out.