Does a ninja have to be a Jōnin to be a Kage?
According to Naruto's case, it might be possible. Hokage is praised to be the strongest shinobi in the village, while strong shinobi is capable of passing Chuunin/Jounin exam. Naruto has only participated in the Chuunin exam twice throughout the whole series. One is during the Chunin Exams Arc, and the other one is in the 5th OVA of Naruto Shippuden. He failed both exams anyways. Nevertheless, he's still strong, but his silliness made him failed the 2nd exam. Shinobi rank might not be the most important fact to consider when electing a new Hokage, they need to be loyal and trustworthy, as Naruto has all of it.