Since the Shi'a believe these 14 persons are chosen by Allah (t.) the answer to why is not trivial. You're asking: Why did Allah choose these individuals and not others. Allahu a'lam.
Shias believe these 13 individuals are the best human beings after the Prophet (s.). You could also ask why was Muhammad (s.) chosen to be the last prophet and not prophet Isa (a.). The short answer is likewise: Muhammad (s.) is better.
I think there is a misunderstanding in the question. The Imamat isn't a monarchy. It's not that the next Imam has to be the first son. In many cases it wasn't. Like the first son of Imam al-Hasan (a.) was not an Imam of the Ahl ul Bayt and neither was the first son of Imam as-Sadiq (a.) (not even the oldest living son).
Side note: In a broader sense of Ahl-ul-Bayt there are more members than the ones mentioned above. For example Zainab bint Ali (a.) also referred to herself and the other captive women (like Ruqayya bin Imam al-Husayn) in Karbala as "Ahlulbayt". But your are right that usually we only mean 13 (or 14 if the prophet himself is included) individuals.