I think the only places I've been asked for a card by non-cavers are White Scar & Banwell Bone Caves
Likewise, and I imagine this is most people's experience from a recreational/ sport caving perspective.
That said, my BCA card goes everywhere with me in my wallet, along with my other club membership cards, and I expect the current one will stay a while longer after it expires...
Regardless of the ups and downs and politics of the BCA, I've always been pleased to carry my card and feel associated with our national body. It's a nice reminder every time you open your wallet, during the grind of life, that there are caves out there and you'll be back down them soon.
I doubt I'll ever even open up the digital card.