The CX ‘concession’ was implemented when the British Cyclo-cross Association was merged/consumed by what was then, the British Cycling Federation – around 2001 IIRC.
Up until then, all cyclo cross was nothing to do with the BCF, and just being a member of the BBCA allowed you to enter any CX race.
One of the points we made when merging was that CX should stay as an entry level sport, and Race Licences would not be needed to enter any race.
I’ve always had a Bronze membership since the merger, and did the National Championships with that. I’m not good enough to enter that now so no idea if you can still enter it with a bronze card.