You are absolutely correct that the BLS publishes CPI, but this still happened in the Greenspan era (and that's what I meant by "under Greenspan"). I remember reading (IIRC, in Bill Fleckenstein's book), that the Boskin commission was not free of interference and/or had an intent going in.
Either way, especially in recent years, it's pretty clear that CPI has been a flawed metric. I remember this episode fondly: https://www.wsj.com/articles/SB10001424052748704893604576199...