@EA_Jason
I tried again and they replied in very bad German and told me they cannot login into my account and EA Desktop is just a beta. Furthermore they told me there is no known problem about the friend list in the EA Desktop app. Yeah, nice...
Sorry, this feels like a absolute clown show. Why am I even reporting an issue and try to help improving the software if the support is just ignoring my request and basically telling me to f*** off and stop annoying the support. Nice experience.
EDIT: Well, just started the EA app another time to check if there's an update just to see this:

How am I supposed to take this software seriously if there are bugs like this? I mean c'mon. I wouldn't even complain if this software wouldn't be mandatory to install/play games included in the Game Pass.