So for all who might still be interested, here is how this played out:
I updated my OS to Catalina. No further, not Big Sur. I wanted to take one step at a time.
After the update I started SYSTM and had the same problems as before.
I deleted it from my Mac and re-installed it. Problems were gone. I could log in, download, play videos, all good.
I had tried this on my old OS as well, to no success, so the OS played SOME role in this.
SUF was still installed, so I started it to see if I can download videos there as well. That failed. After I started SUF, I went back to SYSTM, and I had the same problems as before. I was still logged in, but I could not download videos any more. After an OS reboot, everything in SYSTM looked like I do not have a valid subscription. The app still knew my name and my account, but when I tried to download, it told me I should start the 14 days trial to be allowed to do so. I logged off, and when trying to log back in, that was not possible any more. Back at square one.
So, I deleted SYSTM again, re-installed again, and it worked again.
To summarise all this, I do not think it was primarily an OS problem. The outdated OS played a role in it, I have no idea what was going on there in the background. But even on Catalina I could reproduce the problem. It must have to do with running SYSTM and SUF in parallel.
Anyway, my setup now is stable, at least as long as I don’t touch SUF. I will keep it sitting there on my harddrive for a few more weeks, just to be sure. Come November, if I will face no further issues with SYSTM, I will eventually delete it.
Everyone you supported me here and came up with ideas how to solve my problem, thank you so much!
Great to have such a friendly and helpful community!
May you all suffer well!