Hey, just in case someone else runs into this and it wasn't that their flash drive was dying (mine wasn't), I just happened to click around in the upper right hand corner on the Upgrade License button and saw something about Unraid not officially supporting reverse proxies, so I navigated to my Unraid server GUI via IP address instead, and voila!
Manual Flash Backup now works again normally! I didn't bother to look further into why exactly this caused my issue (same as above, the Unraid logo keeps gyrating but I never get prompt to save file), since the workaround is simple enough for the occasional times I need to do this. Now, I can attempt to upgrade to 6.12.4! I've been putting this off for months due to this little issue.