If Birbal was so smart, why didn't he become a king?

He was a King. Duh.

He was given the title sometime in the 1550s. However he was renowned as a Poet and Admintrator even before this, and had quite a bit of independent wealth from his service in the Court of Rewa as a young man.

At the time of his death, he was one of the leading military commanders of the Empire, despite not being a front-line general. He was most likely primarily responsible for Artillery logistics, & there's some sign that he funded a good bit of the contemporary research on Chemicals in India.