Should I get my dad's birth certificate?

My dad passed away 2019. I was trying to get his birth certificate just to have it cause he is my dad. My dad always told me he was adopted. His adoptive parents name is on his death certificate. However, when I went to the Register of Deeds I was told that they have no record of my father's birth. I presented them my father's Death Certificate still no record of my dad's birth. I told tthe County Courts who my dad's natural mother was. My dad's natural mother was always in my father's life, even after giving him up. The Courts pulled her name up in the system and she had a son born the same day and year as my father with a different name. The Register of Deeds told me that my father was never legally adopted and the name I have always known my father by is not my dad's real name. I was in total shock. I was told to go to Vital Records which is an hours drive from where I live to see if they can retrieve a birth certificate with my dad's name on it that he have always went by. What should I do. The County told me that I would not get anything because my father was never legally adopted. That my father was given away and his named was changed.
Should I get my dad's birth certificate?
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