The OS boots into RAM if there is >=1GB so it's possible to remove boot media and the system will continue to run, but this will make the persistent /storage area unavailable and when it reappears after unclean dismount the kernel will consider the partition to be dirty and remounts it read-only to prevent data corruption issues. This prevents Emby from making ad-hoc saves, and Kodi (Jarvis) only saves it's settings on shutdown, so when you reboot to solve the issue any changes since previous boot are lost.
If it was 1/2 boxes I'd guess dodgy USB ports or USB sticks, but if it's 2/2 and you've already swapped the USB sticks it's either something in hardware like a BIOS or PSU issue that results in interrupted power to the USB ports, or it's software. If it's something in software, it's some kind of kernel level bug that probably doesn't leave much trace: "journalctl --no-pager -b -1" will show the previous boot journal including the diconnect/reconnect, but i'd be surprised if anything is revealed; and I'd be surprised if really was a software issue.
As general advice: I worked for several companies who released products that boot from USB/SD media and a myriad of issues have been miraculously resolved the second they reverted to using old-fashioned spinning rust or SSD's. We don't advocate long-term use of removable media, as while it can be done, the underlying media is not designed for it. I personally consider all USB/SD installations to be temporary. It's never about if things will go wrong, only a matter of when it takes place.