> I know trying to innovate in the public education sector is slow for obvious reasons but shouldn't the private sector be able to scale easier, or at least at a faster pace than it is now.

First we should stop repeating that the private education system is better than the public education system. This happens only where there is corruption and public education system is deliberately stunted to benefit the private education sector.

For what a good education system looks like, one should look at the Scandinavian countries, more specifically Finland. Few years ago I was watching a video on YouTube on Finland's education system. In that a teenage girl, who was preparing for 'teachers exam', was asked on how the exam is, and she replied that it is the toughest, most competitive and prestigious exam. And this is the difference between good public education system and other failed education systems. If the system does not recruit the best and then not provide them with the prestige in the society then the system erodes over time. There use to be a comedy series on TV in the UK where one of the sections of each episode was dedicated to some profession where a character, shown as representing that profession, use to make some funny comment. You know what they showed for a school teacher? They showed a worn down, energy less unshaven skinny guy who, standing in front of the camera, looking emotionless says "I tried to get job in all the other sectors, but I did not get any. So I became a teacher". And the audience laughed! Imagine this is the situation in a developed country, what will be the situation elsewhere.