Your questions touches on a number of issues within the emerging philosophical field of procreative ethics, the field addressing questions concerning the ethics of reproduction and parenting.
I concur with the spirit of your last sentence: It is interesting that landmark legal rulings in the United States establishing legal rights to use birth control and the right to abortion both appealed to the right to privacy. But if there is a right to procreate, it is probably not best modelled on a right to privacy.
Your comments about licensing parents echo a well-known argument given by Hugh LaFollette in a 1980 article (http://www.hughlafollette.com/papers/licensing.parents.pdf). Here's my reconstruction of LaFollette's argument:
1. Incompetent parenting is harmful to children.
2. Societies are justified in restricting access to activities that are potentially harmful to others if those restrictions significantly reduce the likelihood of harm from those activities. (Compare, for example, driving cars or practicing medicine: We license these activities precisely because of their potential for harm.)
3. Licensing would-be parents would significantly reduce the likelihood of harm from incompetent parenting.
Therefore,societies are justified in licensing prospective parents.
LaFollette notes that most societies have very stringent requirements for adopting a child, so if one thinks those requirements are justified (that we should 'license' those who choose to parent children who are already born), it would seem no less justified to require licenses of parents who create their own child.
I won't weigh in with my own appraisal of this argument, leaving it, as they say, as an exercise to the reader. One of the admirable things about LaFollette's article is that it engages with many objections. I'll conclude simply by noting that, in my estimation, the most serious worries concern whether the licensing scheme would violate other rights of prospective parents.