exFAT is just not a terribly reliable filesystem. It has no protection whatsoever against power interruptions or bugs, and it's not native to Linux, really. Its only real strength is interoperability, since all current operating systems can read from and write to it.
Using it as the basis of the Mister distro means that anyone can get it going and load ROMs, no matter what OS they use, so it's an important choice for onboarding newbies and expanding the Mister community. But between FAT and SD cards both being unreliable, it can be a poor choice for longer-term usage.
If you use an outboard hard drive or SSD, you can use any filesystem you like. Linux's native ext4 is a pretty good choice, because it's got a journal, and can thus recover from power failures... usually. It's not perfect, but it's way way better than FAT.
You can also set up a NAS and use any filesystem you like on the server. Ext4 remains a good choice there, too. I'm using ZFS on my server, which is incredibly reliable, but I'm seeing terrible performance at writing hard drive images. The AO486 core, for instance, is glacially slow at writing files to storage. I plan to set up an SSD running ext4 on my server to see if images work better there; I'm suspicious that ZFS's never overwriting blocks is causing the extremely bad performance.
I'd avoid btrfs; they've never really gotten that thing fully reliable. Most folks running desktops don't have too much trouble with it anymore, but I've heard so many horror stories about people who've depended on it, and lost data. ext4 seems much more trustworthy. It's been holding up Linux systems successfully for a long, long, long time.
edit: out of interest, I used Windows to check the Mister drive, and it had "volume bitmap errors", which I think are sectors incorrectly marked as used, with no associated files or directories. There was no data damage, and Windows fixed it fine, but the filesystem was slightly corrupt.