The Imperial Court had its Official Schools.
In terms of culture, iconic institutions like the Imperial College and the Imperial Academy existed, and beneath them, there were also schools, but these were mostly established for the nobility's Disciples, not for the general public.
In the realm of Martial Arts, there were training systems like the Feathered Forest Army which would assimilate promising prospects internally; after all, whether someone had Talent was easy to see, unlike with academics, where someone might seem intelligent yet repeatedly fail to succeed.
Honestly, creating a complete system stretching from elementary school through middle school to university wasn't practical.
There were many reasons.
First and foremost, Martial Arts Schools were typically run by local powers, and meddling with that, cutting into their pie, was likely to be more trouble than it was worth;