Documenting both a relationship and birth fact is certainly not duplication. In my opinion one of the main purposes of genealogical research is to prove the relationship between family members, especially parent / child.
There are many instances were you have proof of birth (date or location) where you do not have proof of the parentage. Evidence of birth does not also equal evidence of both mother(s) or father(s) relationship to the child.
There are different TYPES of relationship to parents (biological, adoptive, step). Different documents pertain to each of these and again in many cases have nothing to do with the birth of the child.
Re the point of the CA death certificate. Only the SURNAME of the deceased persons parents is shown. If the deceased persons came from a family of three boys which of those three boys is the parent of the deceased person. Once you find enough evidence to determine which one is the likely parent you need to source that decision somewhere.
One of the main purposes is showing how you have determined the relationship you have assigned in your database. How will this read in reports or a book you may decide to produce without the ability to show your sources for the relationship you have established?