Is a German birth certificate, the same as citizenship?

The title basically, I am looking into future possibilities of immigrating to different countries. My mom was born in Germany, and has a birth certificate issued from Germany. Since I was young I was told that my mom had dual citizenship.

Because of this I was looking into possible pathways for citizenship by decent for Germany, but I soon found out that my mom may have not had citizenship at all. From what I understand, in Germany to have citizenship you need to be born with atleast one parent with German citizenship to be German citizen. Both of my grandparents on my moms side were born American citizens, but my mom had to naturalize to America once they had moved back.

So this left me with the question that is the title of this post. I want to know if this still a potential pathway that I am eligible to take for future immigration, or is this more complicated?