Before the NHS app came along, all I had was my Public Health England card. I had my first jab at a GP surgery, and they didn't bother to write my name or the date given on the card, just the expiry date of the AZ batch.
When it came to my second one, at a mass vaccination centre, they added my name and this time dated it (although my first date is still missing).
MrsNWI also didn't get her name recorded, and then when attending the same place for her second jab was questioned as to whether or not she was sure it was her card.
Now at any time either of us could have altered that data, or put someone else's name on it, or claimed a second jab before we'd actually received it. It shouldn't exactly be a surprise, in the absence of specific guidance to the contrary, that a similar CDC card was greeted with a degree of suspicion given the risk of BA being fined if they'd incorrectly carried the passenger.
It's not incompetence as was claimed, it's due diligence.