It is quit simple. If it weren't anonymous, many (if not most?) children born this way wouldn't come to live in the first place.
What you are suggesting is in the end this: If children are denied to know about their farther, they shouldn't be conceived. A rather drastic stance, considering that many children grew up to have fine lives without meeting their father.
In fact if this were enacted, I would be the one to rent.
The system in place isn't perfect, but basically nothing is if you want it to work.