We should start from <PERSON>'s answer, which basically states that this already happens today. PPE : The degree that runs Britain. Or 2/3rd Cabinet Privately educated.
As in my comment on <PERSON>'s answer the distinction would be that this isn't society choosing to educate the leaders of tomorrow for their role from a young age, but family and social connections guiding their own youth onto a know successful path.
As these articles demonstrate this can lead to a very narrow range of backgrounds for the leaders of the country. Your two options for why a society decides to recruit 14 years olds specifically into "political academies" are to either re-inforce this narrow world view or to expand it.
Reinforce The current crop of leadership is worried that the masses are getting uppity. They want to make sure that the retain their hold on power without making it too obvious. 714
So the insitute a national program to find and develop the leaders of tomorrow. They produce a standardised test, throw it at the relevant year group and award the top 1% of marks with elite education from 15 to 24, followed by placement into junior roles in government offices with a clear path to high level political staff in either MP/Party roles or Civil Service management tracks. Amazingly this "standardised and fair" test is as easily gamed as the 11+ and leads to the established elite getting their friends and relatives onto the program and a nice cash saving at home, with a few spoilers thrown in to show the system is "open" to everyone.
Expand A maverick political leader from outside the usual channels manages to come to power, perhaps they were the 2nd in command of a "normal" leader as a ticket balance and successfully navigate a succession battle following an unexpected death/scandal. This new leader institutes the same program as above, but their test is focused on apptitudes rather than knowledge and cannot be gamed (for some reason). At 14 it's assumed that teenagers will have a strong connection with their roots, and the post 14 training will emphasise maintaining ties with the source community to retain these diverse viewpoints. 261