DNA testing is very sensitive investigation. It is classified into two. Rest of mind DNA test and Legal DNA test. In rest of mind this where the man felt his not comfortable with the activities of his wife. Therefore choose to clear any doubt without any legal registration. While legal DNA test is where both couple are in dispute. Acquisition and Advice of cheating. Such DNA test is refer as legal. Law have to sign such results.
Making a postnatal Bill for DNA test is optional, it is unacceptable to impose such into ones marriage. Besides the outrageous Bill for DNA test can't be affordable by many. DNA testing is only requested for promiscuous marriage.
This is true. If you are in doubt, you won't be at peace with yourself up until you did this step. This is also a very solid ground for Legal issues if both parties are in conflict. But for regular deliveries without any issue, I don't think you need to include this in the service bill because this is quite expensive. And it is not very practical especially if the family can't afford such service. Unless, we have this test kit where you can get the results in just few hours at a very cheap rate.
But if you trust your spouse enough, for me, there's no need for this and so this is only additional bill for the family if it will be like a mandatory test after birth. For me, it should be an optional and not mandatory.
in the meantime, in the US -

So I don't know in other countries. There may be good reasons like having database so it is easier for criminal investigations and such. But we all know that not all people can afford this. So for me, this is fine as long as it is not mandatory.