Hi,
Ok, just did a flight without loading the flight plan via MSFS (I sometimes do that so I can get the route already loaded)...so entered entire flight plan manually on FMC and indeed all worked well.
So there is probably some data issue on the pln file downloaded from Simbrief downloader or on how this gets translated into the FMC of the FBW A320.
But indeed entering the full flight plan manually into the FMC fixed the problem 🙂