In the UK, it’s not an absolute no, but a “be careful”. Interestingly, my wife’s sleep habits changed considerably. She was instinctively aware of where our baby was, even while asleep.
The main dangers seem to be either the dad (my instincts were far less affected) or a sedated mum. It also becomes a lot less risky when the baby can move. Our daughter was perfectly capable of making her comfort concerns felt.
It’s not zero risk, but it’s far lower than you might think. New mother sleep deprivation is quite different to normal sleep deprivation. I see why the default advice is what it is, however.