Installed Fedora template, created all Fedora based system qubes and set them to work. It only got worse… 
The stick either constantly appears in storage mode and then immediately disappears or rareley even doesn’t appear initially. Even usb-modeswitch configurations don’t help. I just wasted two days!
Can try also Debian 12 template since the stick showed better work with this OS but soon or later Debian 12 reaches its EOL date so there will be no updates. It’s sad. 
Or maybe the Debian 13 devs will release update that will finally fix this issue and I can return to Debian 13. Though since this problem is even worse on Fedora then it means that there is something wrong with interaction with this stick in all latest Linux distros