This idea that bloodlines are relevant is anachronistic superstition. Several of my friends are not genetically related to their children. It's simply not relevant.
Technically this is appeal to tradition. Certainly it's true that our various worldly cultures are strongly influenced by this idea, but factually speaking... before paternity tests, there was literally no way to know for men if a child was their paternal child, especially in societies with very little physical attribute variance. Therefore , it's not really true that "it's the most important thing," because we've never been able to tell for certainty who a child's father is until very recently. Even with physical attributes, our knowledge of genetics is relatively recent in human history, so there's probably lots of false positives / negatives (with hair or eye color for example).
> What if someone just dropped off a random child at your doorstep and tells you you’re responsible for its upbringing now.
This doesn't make sense in the context, is this a red herring or a strawman or similar? I believe the discussions is around situations that would normally involve a paternity test, which to me all essentially double checking whether a woman could have been impregnated by some other person than the expectant father.
They are not remotely the same.
Legally it might not be relevant for whatever structural reason, but the reality is that it matters a lot.