I reinstalled again yesterday evening and this morning I am finally able to progress into Mission 2.
I did not do anything different but if i saw it correctly - today I Updated to Vers 1.05 and before i think i had version 1.04.
So maybe the latest Patch fixed the Problem.
Thx to all you guys again for helping and brainstorming.
Obviously this bug/problem is not super common, but rather rare - so here is a List of possible Solutions/Fixes, if u stumble across this.
Some of those worked for some people, but as I am not sure what exactly worked for me in the end, or if the Patch might fix the bug, i cannot guarantee that any of this will work.
1. Make sure the game is fully installed (35,09 GB) - it is actually startable after Downloading 13 GB, and wont give you a DL-Indicator after that point - only by going to the Information-tab on the Home-Screen you can see the actual Size/DL-Progress.
2. If u played the Demo there seems to be a Problem with the Saves-files. Same if u switched Consoles. In this case it might seem to work to update Online-Save-files to progress (I didn't play the Demo nor do i own PS-Plus, so I just read about this).
3. Reinstall again and again and again and again..... (This worked for me - but i didnt change anything in between)
3.5. Some People reported that it just started working at somepoint without them doing anything, i would guess so here applies point 1.
If anyone finds any other method while/after having this issue, ill add it to the list.