The file inside my documents and the fix above didn't work for me either but this one did so worth a shot.

Inside your \Program Files (x86)\Square Enix\Final Fantasy XIV - A Realm Reborn\boot is a .ver file called ffxivboot. Take that file and drag it onto your desktop, open it with notepad, there'll only be one string of numbers, change the last number from 1 to 0. Save the file, and drag it back into folder. Then run the ffxivboot.exe file as an admin, and your launcher should update.

For steam it's inside the steamapp library where you installed the game
For me - D:\SteamLibrary\steamapps\common\FINAL FANTASY XIV - A Realm Reborn\boot