Turns out the packages from the official repositories do the trick, as long as you do a restart after installing them. :)
Install exfat-utils and exfat-fuse packages from the official repos:
sudo apt-get install exfat-utils exfat-fuse
Reboot your computer:
After reboot you will have a working exfat file system (read and write support, but not formatting the drives with exfat via Gnome Disks and GParted).
Thank you for all the suggestions, especially the PPA from
relan, but the packages in that PPA are older than the ones in the official repositories. This might change when updates start coming in, as the PPA might include new releases more quickly.