So I would think that the new law is actually making things much more clear for the situation you just mentioned, rather than more ambiguous and problematic (due to the existing laws and customs you are speaking of, which can be more subjective and therefore lead to the problems you mention).
If there were any issues (e.g. if someone called the police), the person in that situation could just show a picture of the birth certificate on their phone, which would clarify any possible confusion and would avoid getting into trouble, wouldn't it?
In my view, leaving things the way they are, or making things more subjective only serves to add to the confusion and/or add problems, rather than making things more clear (especially when we're talking about the law).
That said, if you are going to have separate bathrooms (and therefore categorize people), I don't think the birth certificate is the right way to do it, if it can't be changed post-birth.