Hi again everyone,
TLDR: It was the flash media - they were too old. Using a newer flash drive solved the issue.
Details:
After trying and failing the Fedora install using one of the two media I've been using (a novelty 8gb flash drive from 2014, and a 32bg SanDisc Cruzer Blade from about two-three years ago), I opened a support ticket with System76.
Their support representative asked me by email what media and ports I was using.
The Thelio Mira b4 comes with USB-3.0 and USB-4.0 type-A ports, and one type-C port.
I have one extra flash drive with both type-A and type-C connectors, so I used it to try the type-C port - and that booted properly.
I next tried the same media on one of the USB-3.0/type-A ports, and this one also booted.
According to the System76 rep, any USB-2.0 media should work, as long as it has enough storage for the Live ISO.
I must confess to feeling a little embarrassed at not trying the third flash drive before posting here 🤦
Since the various Live disks did reach the pre-boot menu, I had no reason to thing the media were at fault.
Thanks again to everyone for your insights and suggestions!
- Edit -
Having managed to boot the Solus Plasma Live session, the Calamares installer crashed before it finished the installation. If I can't figure this one out on my own however, I'll start a new thread, as it is not strictly speaking related to booting the Live Disk.