OK, so the terakoya are the older, more traditional schools. I get that.

Honestly, since I'm a lazy-ass, I'll just have them put Kenji in a terakoya since there were more of them in Tokyo and there was probably one near the dojo (where ever the hell Kamiya Dojo is actualy located!).

I don't think Kenshin would approve of the militaristic teachings in the public schools. All those public schools were were military indoctrination institutions. I can't see him and Kaoru wanting Kenji exposed to that.