Thanks everyone for the responses.
Yes, it's macOS and Buster.
I've checked out the issues page, and this catches my eye:
That doesn't bode well, as I'm pretty sure I have a very early ThumbDV.
However, I can say for sure that I have unplugged and reinserted the ThumbDV, which is the suggested workaround for the "hardware issue" that is blamed for the connection to the device being broken when a machine sleeps. In my case, I can't get the software to connect to the device on the *first* occasion.
I did wonder whether new tighter sandboxing (security) restrictions for App Store Mac apps might have left the app unable to negotiate the rights to connect directly to hardware with the OS. However, I think I see other people able to use the device with macOS High Sierra, which probably disproves that theory.
I have also tried setting both available serial speeds, with no luck (I assume there's an attempt to reconnect to the device when the speed is changed - though there's no visible confirmation that the new speed is active). In any case, I've also tried unplugging and reconnecting the hardware with each speed selected.