The private key there is only an implementation detail, though. The reason people don't care about it is because in the case of a SIM card you can get a replace

The private key there is only an implementation detail, though. The reason people don't care about it is because in the case of a SIM card you can get a replacement SIM while keeping the same phone number (which is what people really care about), and in the case of a credit card where a premature replacement might involve a change of credit card number - for one-off payments it doesn't matter anyway, you just start using the new card and that's it - and for recurring payments all businesses already need to allow for updating your credit card details because credit cards regularly expire anyway, so having to possibly manually update all your recurring payments might be a bit tedious and annoying, but it's certainly no impossible, out-of-the-ordinary task