If you are curious about what is revealed to anyone possessing your PNR and surname, just enter them without being logged into your BAEC membership.
You will see a lot of details, including your BAEC membership number and email address.
It's not just BA though, going to MMB on on AY website and entering them will essentially reveal the same.
Everyone should be careful where you expose your PRN.
I would say that the PRN should NOT be printed on the BP for that reason (people leave it behind, drop it, etc. etc.).
Back to the subject of the card, I once held a physical Blue card, and that's far more of a novelty to me than a Gold/GGL/CCR card. In fact I didn't even get my GGL card for the latest membership year for whatever reason and couldn't even be bothered chasing it up (I would chase up the CCR card if it doesn't turn up though, as it does cause inconvenience not to have it).
Now, if they issued me with that GOLD POO card, I will send it back to BA, asking them to present me with something that doesn't look like a poo, since the GGL crest will make it look even more like a poo, just with an extra fly around it.
(Those cards are a bit like the optical illusion - once you see the poo, you always see the poo... Sorry, it's been ruined for me, so I really hope this isn't the real design.)