Ok, I've uninstalled Origin. The uninstall of Origin did remove all my installed Origin Games as well.
I did reinstall it to another path.
My first install path was not the default. Now I did install to the default path.
Changed the Game install path in the settings.
Started the Download of BF5.
After around 25% Download of the 50GB it showed me that I can start the game already. And the Game started. I will leave the game started until the install will be finished. Will see if it will still run and if it will still start after a relaunch.
Crossing fingers...