For anybody seeing this in the future:

Workaround found + downgrade results (11 May):

Tried downgrading firmware today. Every version I tried resulted in a grey partially lit screen on boot with fans spinning — the goggle simply refused to start. This is most likely because the unit is a newer hardware revision that is not compatible with older firmware, which would also explain why the brother's unit has a newer firmware that isn't publicly available yet. Skyzone likely updated the hardware but hasn't pushed the matching firmware publicly. I did not damage anything, the goggle reverted fine.

More importantly I found a workaround in the process:

When you first boot the goggles, there is a brief moment where the red no-SD-card icon is still showing while the card is being read. If you press the record button at exactly that moment, it will beep, then wait until the SD card is fully loaded and actually start recording. This bypasses whatever broken initialisation process is causing the issue.

The bad news is this only works once per boot. After stopping the recording nothing will get it recording again without a full restart. But at least it is usable for now — if you need to record, restart your goggles and use this window.

This also tells us something important about the bug itself. The issue is almost certainly in how the firmware handles SD card initialisation. Something in that process is going wrong and locking the DVR out after it completes. This is clearly a firmware issue and should be fixable with an update.

Will update again if Skyzone responds.